Window Study of Nivolumab With or Without Ipilimumab in Squamous Cell Carcinoma of the Oral Cavity

Inclusion Criteria: * Pathologically confirmed squamous cell carcinoma of the oral cavity. Clinical stage \>=T2 (primary tumor greater than 2 cm in size) and/or evidence of regional nodal involvement by clinical exam or imaging * Only patients 18 years and older are eligible. There is no upper age limit but the patients must be able to medically tolerate the regimen. Adverse event data are currently unavailable on the use immune checkpoint blockade for participants \< 18 years of age, and thus children are excluded from this study * ECOG performance status \<=1 * Patients much be a surgical candidate (e.g. their disease must be considered resectable before any treatment and must have no serious medical contraindications that definitively preclude undergoing general anesthesia) Ability to understand and the willingness to sign a written informed consent document * Women of childbearing potential (WOCBP) must agree to use appropriate method(s) of contraception (see Appendix B). WOCBP should use an adequate method to avoid pregnancy for 23 weeks (30 days plus the time required for nivolumab to undergo five half-lives) after the last dose of investigational drug. WOCBP is defined as any woman or adolescent who has begun menstruation and is not post- menopausal. A post-menopausal woman is defined as a woman who is over the age of 45 and has not had a menstrual period for at least 12 months * Women of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of HCG) within 24 hours prior to the start of nivolumab * Men who are sexually active with WOCBP must agree to use any contraceptive method (see Appendix B) with a failure rate of less than 1% per year. Men receiving nivolumab and who are sexually active with WOCBP will be instructed to adhere to contraception for a period of 31 weeks after the last dose of investigational product Women who are not of childbearing potential (ie, who are postmenopausal or surgically sterile as well as azoospermic men do not require contraception) * Participants must have normal organ and marrow function as defined below: Laboratory parameters: WBC ≥ 2000/uL, Absolute neutrophil count (ANC) ≥ 1500/mm3; Platelets ≥ 100,000/mm3; Hemoglobin (Hgb) ≥ 9 g/dL; Hgb-A1C ≤ 7.5%; Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 × upper limit of normal (ULN); Bilirubin ≤ 2.5 × ULN (≤ 4 × ULN for subjects with Gilbert's disease); Alkaline phosphatase ≤ 2.5 × ULN; Creatinine ≤ 1.5 × ULN Exclusion Criteria: * Pathologically proven, radiologic or clinical evidence of distant metastatic disease (this includes all disease below the clavicles, as well as disease metastatic to the bone, brain, or in the spinal canal) * Any prior immunologic cancer therapy with systemic inhibitors of the PD-1 or CTLA-4 pathway * 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 * Individuals with a history of a different malignancy are ineligible except for the following circumstances: if they have been disease-free for at least 2 years and are deemed by the investigator to be at low risk for recurrence of that malignancy; or if diagnosed and treated within the past 2 years for cervical cancer in situ or basal cell or squamous cell carcinoma of the skin * Prior radiation to the head and neck region * Prior chemotherapy within the last 2 years * History of pneumonitis or interstitial lung disease * Has evidence of active, noninfectious pneumonitis that required treatment with steroids. * Active, suspected or prior documented autoimmune disease that has required systemic treatment in the last 2 years with immune modifying agents (e.g. replacement therapy such as thyroxine, insulin or physiologic corticosteroids is not an exclusion criteria). This does not include patients with vitiligo or type 1 diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger * The subject is known to be positive for the human immunodeficiency virus (HIV), HepBsAg, or HCV RNA * Lack of availability for follow up assessments * Concurrent administration of other cancer specific therapy during the course of this study is not allowed * Patients who require systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \> 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease * History of allergy to study drug components * History of severe hypersensitivity reaction to any monoclonal antibody * The investigator's belief that the subject is medically unfit to receive nivolumab, and/or ipilimumab or unsuitable for any other reason * Has received a live vaccine within 28 days of planned start of study therapy