Nivolumab, Ipilimumab, and Radiation Therapy in Treating Patients With Stage III-IVB Head and Neck Cancer

Inclusion Criteria: * Patients must be 18 years of age and older * Pathologically confirmed squamous cell carcinoma of the head and neck (SCCHN), not previously treated * Stage III-IVB disease of 1) oral cavity, 2) HPV-negative (p16-) oropharynx, 3) larynx, 4) hypopharynx * Oropharyngeal primaries that are HPV-mediated (p16+) must be stage II-III. Stage II pateints must be either N2 or, if T3N0 or T3N1 they must also have at least 20 pack year history of smoking * Tumor sample must be available for HPV p16 and PD-L1 testing * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * White blood cells 2000/ul or more * Absolute neutrophil count 1500/ul or more * Platelets 100,000/ul or more * Hemoglobin 9 g/dl or more * Bilirubin less than or equal to 1.5 x the upper limit of normal (except subjects with Gilbert syndrome, who can have total bilirubin \< 3 mg/dl) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 3 x the upper limit of normal * Glomerular filtration rate (GFR) greater than or equal to 40 ml/min using the Cockcroft-Gault formula or Serum creatinine less than or equal to 1.5 x upper limit of normal (ULN) * Women of reproductive potential should have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 24 hours of the start of study drugs * Women of reproductive potential must use highly effective contraception methods to avoid pregnancy for 23 weeks after the last dose of study drugs; "women of reproductive potential" is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal; menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes; in addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level more than 40 mIU/mL * Men of reproductive potential who are sexually active with women of reproductive potential must use any contraceptive method with a failure rate of less than 1% per year; men who are receiving the study medications will be instructed to adhere to contraception for 31 weeks after the last dose of study drugs; men who are azoospermic do not require contraception * Informed consent: all subjects must be able to comprehend and sign a written informed consent document Exclusion Criteria: * Primary nasopharyngeal carcinoma * Patients with brain metastases * Patients who have participated in a study with an investigational agent or device within 2 weeks of initiation of treatment * Any prior radiotherapy to the neck * Patients with known contraindications to radiotherapy, including inherited syndromes associated with hypersensitivity to ionizing radiation (e.g., Ataxia-Telangiectasia, Nijmegen Breakage Syndrome) * Any prior chemotherapy or radiation therapy for the current diagnosis * Any prior therapy with anti-PD-1, anti-PD-L2, anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell co-stimulation or immune checkpoint pathways * Any history of a sever hypersensitivity reaction to any monoclonal antibody * Any history of allergy to the study drug components * Any concurrent malignancies- exceptions include- basal cell carcinoma of the skin, squamous cell carcinoma of the skin, superficial bladder cancer or in situ cervical cancer that has undergone potentially curative therapy; patients with a history of other prior malignancy must have been treated with curative intent and must have remained disease-free for 3 years post-diagnosis * Any diagnosis of immunodeficiency or current immunosuppressive therapy including \>10mg/day of prednisone within 14 days of enrollment is not permitted * Patients that have an active autoimmune disease requiring systemic treatment within the past 3 months or a documented history of clinically severe autoimmune disease, or a syndrome that requires systemic steroids (\> 10 mg daily prednisone equivalents) or immunosuppressive agents; subjects with vitiligo, type I diabetes mellitus, or resolved childhood asthma/atopy would be an exception to this rule. Inhaled or topical steroids, and adrenal replacement steroids \</=10mg daily prednisone equivalent, are permitted in the absence of active autoimmune disease. Subjects with hypothyroidism stable on hormone replacement or Sjorgen's syndrome will not be excluded from the study * Patients with evidence of interstitial lung disease or active, non-infectious pneumonitis * Patients with a known human immunodeficiency virus infection (HIV 1/2 antibodies) or acquired immunodeficiency syndrome (HIV/AIDS), active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected) * Patients who have received a live vaccine within 30 days prior to the radiation therapy * Patients must not be receiving any other investigational agents * Patients with uncontrolled intercurrent illnesses including, but not limited to an active infection requiring systemic therapy or a known psychiatric or substance abuse disorder(s) that would interfere with cooperation with the requirements of the trial * Women must not be pregnant (as above) or breastfeeding