Inclusion Criteria:

          -  American Joint Committee on Cancer (AJCC) stage III/IVB, excluding T1N1,
             histologically or cytologically confirmed squamous cell carcinoma or undifferentiated
             carcinoma of the head and neck; patients should not have distant metastasis; primary
             sites include: oropharynx, hypopharynx, larynx

          -  Patients must have high or intermediate risk disease, defined as follows:

               -  High risk: non-oropharyngeal subsite including larynx or hypopharynx (p16 status
                  not required) or human papilloma virus (HPV)/p16- oropharynx subsite

               -  Intermediate risk: HPV/p16+ oropharyngeal squamous cell cancer with: >= 10 pack
                  (pk)-year (yr) smoking history and >= N2 nodal disease, or the presence of T4
                  tumor or N3 nodal disease, irrespective of smoking status

          -  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

          -  Patients should be newly diagnosed HNSCC, with no prior therapy for this disease

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

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,200/mcL

          -  Platelets >= 75,000/mcL

          -  Total bilirubin =< 2 mg/dL (=< 3 mg/dL in case of Gilbert's syndrome)

          -  Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 2 times
             institutional upper limit of normal (IULN)

          -  Creatinine clearance >= 40 mL/min/1.73 m^2

          -  Patients must have the ability to understand and to sign written informed consent

        Exclusion Criteria:

          -  Patients who have had prior chemotherapy, radiotherapy, or surgery with curative
             intent for HNSCC

          -  Patients with a history of prior treatment with ipilimumab, anti-programmed cell death
             1 (PD 1) antibody, cluster of differentiation 137 (CD137) agonist or other immune
             activating therapy such as anti-cluster of differentiation 40 (CD 40) antibody

          -  Patients who are receiving any other investigational agents

          -  Autoimmune disease: patients with a history of inflammatory bowel disease, including
             ulcerative colitis and Crohn's disease, are excluded from this study, as are patients
             with a history of symptomatic non-gastrointestinal autoimmune disease (e.g.,
             rheumatoid arthritis, systemic progressive sclerosis [scleroderma], systemic lupus
             erythematosus, autoimmune vasculitis [e.g., Wegener's granulomatosis]); central
             nervous system (CNS) or motor neuropathy considered of autoimmune origin (e.g.
             Guillain-Barre syndrome and myasthenia gravis, multiple sclerosis)

          -  Patients with known immunodeficiency disorder, or presumed to be unable to respond to
             anti-cytotoxic T-lymphocyte-associated protein 4 (CTLA 4) monoclonal antibody (mAb)

          -  Patients with distant metastatic disease (stage IVC)

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to cetuximab or ipilimumab

          -  Patient is < 2 years free from a second primary malignancy unless the other malignancy
             is non-melanomatous skin cancer or an in-situ tumor treated with curative intent

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements; patients with chronic hepatitis B or hepatitis C infections are
             excluded

          -  Pregnant women are excluded from this study

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible