Ipilimumab, Cetuximab, and Intensity-Modulated Radiation Therapy in Treating Patients With Previously Untreated Stage III-IVB Head and Neck Cancer

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