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…