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