Inclusion Criteria:

          1. Participants with recurrent or progressive high-grade gliomas (HGG) (World Health
             Organization (WHO) grade III or grade IV) who are candidates for surgical tumor
             debulking will be enrolled in this trial

          2. All assessments are to occur within 14 days of registration except where otherwise
             noted. The participant and their legal parent/guardian must be thoroughly informed
             about all aspects of the study, including the study visit schedule and required
             evaluations and all regulatory requirements for informed consent. The written informed
             consent must be obtained from the participant and legal parent/guardian prior to
             enrollment

          3. Have a history of previously treated histologically confirmed World Health
             Organization grade III or IV HGG. Previous first line therapy with radiation and/or
             chemotherapy

          4. Have evidence of recurrence or progression of disease by MRI scan

          5. Participants must be adequate medical candidates for surgical resection. The intent of
             surgical resection is to allow both cytoreduction and tumor debulking as part of
             standard of care, and also collect a minimum of 100 mg of tumor tissue for the study
             tissue endpoints

          6. A primary goal of surgery must be cytoreduction, and not solely on diagnostic biopsy

          7. Age: Participants must be > 6 months and < 22 years of age at time of enrollment

          8. Karnofsky >= 50 for participants > 16 years of age and Lansky >= 50 for participants
             =< 16 years of age. Participants who are unable to walk because of paralysis, but who
             are up in a wheelchair, will be considered ambulatory for the purpose of assessing the
             performance score

          9. Prior Therapy: Participants must have fully recovered from the acute toxic effects of
             all prior anti-cancer therapy and must meet the following minimum duration from prior
             anti-cancer directed therapy prior to enrollment. If after the required timeframe, the
             defined eligibility criteria are met, e.g. blood count criteria, the patient is
             considered to have recovered adequately

               -  Cytotoxic chemotherapy or other anti-cancer agents known to be myelosuppressive.
                  At least 21 days after the last dose of cytotoxic or myelosuppressive
                  chemotherapy (42 days if prior nitrosourea)

               -  An interval of at least 12 weeks from the completion of radiation therapy to
                  registration unless there is unequivocal histologic confirmation of tumor
                  progression

               -  Hematopoietic growth factors: At least 14 days after the last dose of a
                  long-acting growth factor (e.g. pegfilgrastim) or 7 days for short acting growth
                  factor. For agents that have known adverse events occurring beyond 7 days after
                  administration, this period must be extended beyond the time during which adverse
                  events are known to occur. The duration of this interval must be discussed with
                  the study chair

               -  Anti-cancer agents not known to be myelosuppressive (e.g. not associated with
                  reduced platelet or absolute neutrophil count [ANC] counts): At least 7 days
                  after the last dose of agent

               -  Interleukins, interferons and cytokines (other than hematopoietic growth
                  factors): >= 21 days after the completion of interleukins, interferon or
                  cytokines (other than hematopoietic growth factors)

               -  Antibodies: >= 21 days must have elapsed from infusion of last dose of antibody,
                  and toxicity related to prior antibody therapy must be recovered to grade =< 1

               -  Participants must not have received prior exposure to PD-1, PD-L1 or CTLA4
                  inhibitors

               -  Stem cell infusion (with or without total-body irradiation (TBI)):

                    -  Autologous stem cell infusion including boost infusion: >= 42 days

         10. Participants must be willing to forego cytotoxic anti-tumor therapies except
             study-defined therapy while being treated on study

         11. Organ Function Requirements:

               -  Peripheral absolute neutrophil count (ANC) >= 1000/mm^3

               -  Platelet count >= 100,000/mm^3

               -  Creatinine clearance or radioisotope glomerular filtration rate (GFR) >= 70
                  mL/min/1.73 m^2 OR a serum creatinine based on age/gender as follows:

                    -  Age: Maximum Serum Creatinine (mg/dL)

                    -  6 months to < 3 years: 0.6 (male and female)

                    -  3 to < 6 years: 0.8 (male and female)

                    -  6 to < 10 years: 1 (male and female)

                    -  10 to < 13 years: 1.2 (male and female)

                    -  13 to < 16 years: 1.5 (male), 1.4 (female)

                    -  >= 16 years: 1.7 (male), 1.4 (female)

               -  Bilirubin (sum of conjugated and unconjugated) =< 1.5 x upper limit of normal
                  (ULN) for age (except participants with Gilbert syndrome who must have a total
                  bilirubin level of < 3.0

               -  Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 3.0 x ULN

               -  Serum albumin >= 2

         12. Pregnancy: The effects of nivolumab and ipilimumab on the developing human fetus are
             unknown. For this reason women of child-bearing potential must agree to use adequate
             contraception (hormonal or barrier method of birth control; abstinence) prior to study
             entry, for the duration of study participation and 5 months after completion of
             therapy. Should a woman become pregnant or suspect she is pregnant while she or her
             partner is participating in this study, she should inform her treating physician
             immediately

         13. MRI within 28 days prior to registration

        Exclusion Criteria:

          1. Current or planned participation in a study of an investigational agent or using an
             investigational device

          2. Has a diagnosis of immunodeficiency

          3. Has tumor primarily localized to the brainstem or spinal cord

          4. Has presence of diffuse leptomeningeal disease or extracranial disease

          5. Has received systemic immunosuppressive treatments (such as methotrexate, chloroquine,
             azathioprine, etc.), aside from anti-neoplastic chemotherapy or systemic
             corticosteroids within six months of registration

          6. Participants with a concurrent condition requiring systemic treatment with either
             corticosteroids (> 0.25 mg/kg daily prednisone equivalent) or other immunosuppressive
             medications within 14 days of start of study treatment. Inhaled or topical steroids,
             and adrenal replacement steroid doses > 0.25 mg/kg daily prednisone equivalent, are
             permitted in the absence of active autoimmune disease

          7. Unable to taper steroids due to ongoing mass effect; a maximum dexamethasone dose of
             0.1 mg/kg/day is allowed, but preferably have been discontinued (inhaled or topical
             use of steroids is allowed)

          8. Has a known history of active TB (Bacillus tuberculosis)

          9. Has a known additional malignancy that is progressing or requires active treatment
             within 3 years of registration. Exceptions include basal cell carcinoma of the skin,
             squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone
             potentially curative therapy

         10. Has active autoimmune disease that has required systemic treatment in the past 2 years
             (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive
             drugs). Replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid
             replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a
             form of systemic treatment

         11. Has known history of, or any evidence of active non-infectious pneumonitis

         12. Has an active infection requiring systemic therapy

         13. Has a known hypersensitivity to any of the study therapy products

         14. Has a known history of positive test for human immunodeficiency virus (HIV) or known
             acquired immunodeficiency syndrome (AIDS)

               -  NOTE: Testing for HIV must be performed at sites where mandated locally

         15. Any prior positive test result for hepatitis B virus or hepatitis C virus indicating
             presence of virus, e.g., hepatitis B surface antigen (HBsAg, Australia antigen)
             positive, or hepatitis C antibody (anti-HCV) positive (except if HCV-ribonucleic acid
             (RNA) negative)

         16. Participants who have had prior allogenic hematopoietic stem cell transplant (HSCT)

         17. Any serious or uncontrolled medical disorder that, in the opinion of the investigator
             may increase the risk associated with study participation or study drug
             administration, impair the ability of the participant to receive protocol therapy or
             interfere with interpretation of study results