Inclusion Criteria:

          -  PHASE I SUBJECTS ONLY: Prior therapy is allowed; for patients enrolled in the Phase I
             portion of the study, patients may have received any number of prior lines of therapy
             including treatment with a BRAF and/or MEK inhibitor; prior navitoclax use will not be
             allowed, unless the patient received < 7 days of navitoclax lead-in on this or another
             study and had to stop for reasons other than toxicity or disease progression

          -  Patients must have histologically confirmed, BRAF-mutant (V600E/K) melanoma
             (molecularly confirmed using validated, commercially available assay performed in a
             Clinical Laboratory Improvement Act [CLIA]-approved laboratory) that is metastatic or
             unresectable and for which standard curative measures do not exist or are no longer
             effective

               -  If test at CLIA-certified lab used a non-Food and Drug Administration (FDA)
                  approved method, information about the assay must be provided; (FDA approved
                  tests for BRAF V600 mutations in melanoma include: THxID BRAF Detection Kit and
                  Cobas 4800 BRAF V600 Mutation Test)

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as >= 20 mm with conventional
             techniques or as >= 10 mm with spiral computed tomography (CT) scan, magnetic
             resonance imaging (MRI), or calipers by clinical exam

          -  Prior therapy is allowed; for patients enrolled in the Phase II portion of the study,
             patients may have received prior immunotherapy (including high-dose IL-2, ipilimumab,
             nivolumab, and other anti-PD1/PDL1 antibodies) or chemotherapy; however prior
             navitoclax, BRAF inhibitor and/or MEK inhibitor therapy will not be allowed

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky >= 70%)

          -  Life expectancy of greater than 3 months

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1 x 10^9/L

          -  Hemoglobin >= 9 g/dl (patients may be transfused to this level)

          -  Platelets >= 100,000/mcL

          -  Total bilirubin =< 1.5 x institutional upper limit of normal OR > 1.5 x institutional
             upper limit of normal allowed if direct bilirubin is within normal range

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 x institutional upper limit of normal

          -  Prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin
             time (PTT) < 1.3 x upper limit of normal (ULN)

          -  Serum creatinine =< 1.5 mg/dL OR creatinine clearance >= 50 mL/min/1.73 m^2

          -  Left ventricular ejection fraction >= institutional lower limit of normal (LLN) by
             echocardiogram (ECHO)

          -  Patients must have a corrected QT (QTc) interval of less than 480 msec

          -  The effects of navitoclax, dabrafenib, and trametinib on the developing human fetus
             are unknown; for this reason, women of child-bearing potential and men with partners
             of childbearing potential must agree to use adequate contraception (barrier method of
             birth control, or abstinence; hormonal contraception is not allowed due to drug-drug
             interactions which can render hormonal contraceptives ineffective) prior to study
             entry, for the duration of study participation, and for 4 months after completion of
             study drug administration; should a woman become pregnant or suspect she is pregnant
             while she is participating in this study, she should inform her treating physician
             immediately; based on studies in animals, it is also known that dabrafenib may cause
             damage to the tissue that makes sperm; this may cause sperm to be abnormal in shape
             and size and could lead to infertility, which may be irreversible; safety and efficacy
             of the combination of dabrafenib and trametinib in pediatric populations have not been
             investigated; dabrafenib or trametinib-dabrafenib combination should not be
             administered to pediatric populations outside clinical trials

          -  Able to swallow and retain oral medication and must not have any clinically
             significant gastrointestinal abnormalities that may alter absorption such as
             malabsorption syndrome or major resection of the stomach or bowels

          -  Ability to understand and the willingness to sign a written informed consent document;
             if a patient has impaired decision-making capacity, a legally authorized
             representative, patients will be allowed to participate

        Exclusion Criteria:

          -  PHASE I SUBJECTS ONLY: Patients must not have received prior navitoclax, unless the
             patient received < 7 days of navitoclax lead-in on this or another study and had to
             stop for reasons other than toxicity or disease progression

          -  Patients who have had immunotherapy, chemotherapy or radiotherapy within 14 days prior
             to the first dose of navitoclax, or prior systemic anti-cancer therapy (chemotherapy
             with delayed toxicity, extensive radiation therapy, immunotherapy, biologic therapy,
             or vaccine therapy) within the last 3 weeks prior to first dose of dabrafenib and/or
             trametinib; chemotherapy regimens without delayed toxicity within the last 2 weeks
             preceding the first dose of study treatment; biologics will not be allowed within 30
             days prior to, or during, navitoclax administration

          -  Prior navitoclax, BRAF inhibitor, and MEK inhibitor is prohibited; (exceptions for
             Phase I are described above)

          -  Patients who are receiving any other investigational agents have received any other
             investigational drugs within 28 days (or five half-lives, whichever is shorter; with a
             minimum of 14 days from the last dose) preceding the first dose of study treatment and
             during the study

          -  Patients must have no clinical evidence of leptomeningeal or brain metastasis causing
             spinal cord compression that are symptomatic or untreated or not stable for >= 3
             months (must be documented by imaging) or requiring corticosteroids; subjects on a
             stable dose of corticosteroids > 1 month or who have been off of corticosteroids for
             at least 2 weeks can be enrolled with approval of the Cancer Therapy Evaluation
             Program (CTEP) medical monitor; subjects must also be off of enzyme-inducing
             anticonvulsants for > 4 weeks

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to navitoclax, dabrafenib, or trametinib, or excipients or to dimethyl
             sulfoxide (DMSO)

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             serious infection, symptomatic congestive heart failure, unstable angina pectoris,
             cardiac arrhythmia, uncontrolled diabetes, or psychiatric illness/social situations
             that would limit compliance with study requirements

          -  Pregnant women are excluded from this study because navitoclax, dabrafenib, and
             trametinib may have teratogenic or abortifacient effects; because there is an unknown
             but potential risk for adverse events in nursing infants secondary to treatment of the
             mother with the study drugs, breastfeeding should be discontinued if the mother is
             treated with the study drugs

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy that predict to interact with any of the study drugs are ineligible because of
             the potential for pharmacokinetic interactions with the study drugs; appropriate
             studies will be undertaken in patients receiving combination antiretroviral therapy
             when indicated; it is not necessary to conduct HIV testing at screening; patients who
             are HIV-positive with undetectable viral loads, not on interacting antiretroviral
             therapy, and have CD4 counts above 300/mm^3 may be eligible after discussion with the
             principal investigator

          -  History of another malignancy; exception: patients who have been disease-free for 3
             years (depending upon tumor type studied or clinical setting, 3 or 5 years can be
             used; e.g., for advanced melanoma and pancreatic studies 3 years is more appropriate
             due to aggressiveness of the disease, while 5 years can be more appropriate for
             prostate or ovarian cancer or adjuvant setting when life expectancy is longer), or
             patients with a history of completely resected non-melanoma skin cancer and/or
             patients with indolent secondary malignancies, are eligible; consult the CTEP medical
             monitor if unsure whether second malignancies meet the requirements specified above;
             exception: patients with history of RAS mutation-positive tumors are not eligible
             regardless of interval from the current study; prospective RAS testing is not
             required; however, if the results of previous RAS testing are known, they must be used
             in assessing eligibility

          -  History of interstitial lung disease or pneumonitis

          -  History or current evidence/risk of retinal vein occlusion (RVO) or central serous
             retinopathy (CSR):

               -  History of RVO or CSR, or predisposing factors to RVO or CSR (e.g., uncontrolled
                  glaucoma or ocular hypertension, uncontrolled systemic disease such as
                  hypertension, diabetes mellitus, or history of hyperviscosity or
                  hypercoagulability syndromes)

               -  Visible retinal pathology as assessed by ophthalmic exam that is considered a
                  risk factor for RVO or CSR such as evidence of new optic disc cupping, evidence
                  of new visual field defects, and intraocular pressure > 21 mmHg

          -  History or evidence of cardiovascular risk including any of the following:

               -  A QT interval corrected for heart rate using the Bazett's formula QTcB >= 480
                  msec on screening electrocardiography (ECG)

               -  History or evidence of current clinically significant uncontrolled arrhythmias
                  (exception: patients with controlled atrial fibrillation for > 30 days prior to
                  randomization are eligible)

               -  History of acute coronary syndromes (including myocardial infarction and unstable
                  angina), coronary angioplasty, or stenting within 6 months prior to randomization

               -  History or evidence of current >= class II congestive heart failure as defined by
                  the New York Heart Association (NYHA) functional classification system

               -  Treatment-refractory hypertension defined as a blood pressure of systolic > 140
                  mmHg and/or diastolic > 90 mmHg which cannot be controlled by anti-hypertensive
                  therapy

               -  Abnormal cardiac valve morphology (>= grade 2) documented by echocardiogram
                  (subjects with grade 1 abnormalities [i.e., mild regurgitation/stenosis] can be
                  entered on study); subjects with moderate valvular thickening should not be
                  entered on study

          -  Known history of hepatitis B virus (HBV), or hepatitis C virus (HCV) infection
             (patients with chronic or cleared HBV and HCV infection are eligible); it is not
             necessary to conduct HBV and HCV testing at screening

          -  Subject has an underlying condition predisposing them to bleeding or currently
             exhibits signs of clinically significant bleeding

          -  Subject has a recent history of non-chemotherapy-induced thrombocytopenic-associated
             bleeding within 1 year prior to the first dose of study drug

          -  A history of glucose-6-phosphate dehydrogenase (G6PD) deficiency

          -  Unresolved toxicity of National Cancer Institute Common Terminology Criteria for
             Adverse Events, version 5.0 (NCI CTCAE v. 5.0) grade 2 or higher from previous
             anti-cancer therapy, except alopecia or an endocrine toxicity related to immunotherapy
             (e.g. thyroiditis/hypothyroidism, adrenal insufficiency, hypophysitis) requiring
             replacement therapy, at the time of randomization

          -  Due to the expected dose-limiting toxicity of thrombocytopenia, the following
             concomitant medications are not allowed during navitoclax administration: clopidogrel,
             ibuprofen, tirofiban, warfarin, and other anticoagulants, drugs, or herbal supplements
             that affect platelet function are excluded, with the exception of low-dose
             anticoagulation medications (such as heparin) that are used to maintain the patency of
             a central intravenous catheter; aspirin will not be allowed within 7 days prior to the
             first dose of navitoclax or during navitoclax administration; however, subjects who
             have previously received aspirin therapy for thrombosis prevention may resume a low
             dose (i.e., maximum 100 mg QD) of aspirin if platelet counts are stable (>=
             50,000/mm^3) through 6 weeks of navitoclax administration; all decisions regarding
             treatment with aspirin therapy will be determined by the investigator in conjunction
             with the medical monitor

          -  Current use of a prohibited medication; the following medications or non-drug
             therapies are prohibited:

               -  Other anti-cancer therapy while on study treatment; (note: megestrol [Megace] if
                  used as an appetite stimulant is allowed)

               -  Concurrent treatment with bisphosphonates is permitted; however, treatment must
                  be initiated prior to the first dose of study therapy; prophylactic use of
                  bisphosphonates in patients without bone disease is not permitted, except for the
                  treatment of osteoporosis

               -  Because the composition, pharmacokinetics (PK), and metabolism of many herbal
                  supplements are unknown, the concurrent use of all herbal supplements is
                  prohibited during the study (including, but not limited to, St. John's wort,
                  kava, ephedra [ma huang], ginkgo biloba, dehydroepiandrosterone [DHEA], yohimbe,
                  saw palmetto, or ginseng)

               -  Anticoagulants or antiplatelet agents except for low-dose, aspirin

          -  Preclinical studies indicate that navitoclax is metabolized by CYP3A4, is a moderate
             inhibitor of CYP2C8, and is a strong inhibitor of CYP2C9; there is also evidence of
             interactions with dabrafenib; therefore, caution should be exercised when dosing
             navitoclax concurrently with CYP2C8 and CYP2C9 substrates; common CYP2C8 substrates
             include paclitaxel, statins, and glitazones, whereas CYP2C9 substrates include
             phenytoin; when possible, investigators should switch to alternative medications or
             monitor the patients closely; CYP3A inhibitors such as ketoconazole and clarithromycin
             are not allowed 7 days prior to the first dose of navitoclax or during navitoclax
             administration

          -  Patients receiving any medications or substances that are strong inhibitors or
             inducers of CYP3A or CYP2C8 are ineligible; current use of, or intended ongoing
             treatment with: herbal remedies (e.g., St. John's wort), or strong inhibitors or
             inducers of permeability-glycoprotein (Pgp) or breast cancer resistance protein 1
             (Bcrp1) should also be excluded; below are a few examples of the agents:

               -  Strong inducers of CYP3A or CYP2C8, since concentrations of dabrafenib may be
                  decreased:

                    -  Antibiotics: rifamycin class agents (e.g., rifampin, rifabutin, rifapentine)

                    -  Anticonvulsants: carbamazepine, oxcarbazepine, phenobarbital, phenytoin,
                       s-mephenytoin

                    -  Miscellaneous: bosentan, St. John's wort

               -  Strong inhibitors of CYP3A or CYP2C8 since concentrations of dabrafenib may be
                  increased:

                    -  Antibiotics: clarithromycin, telithromycin, troleandomycin

                    -  Antidepressants: nefazodone

                    -  Antifungals: itraconazole, ketoconazole, posaconazole, voriconazole

                    -  Hyperlipidemia: gemfibrozil

                    -  Antiretroviral: ritonavir, saquinavir, atazanavir

                    -  Miscellane