Study of Nivolumab and Ipilimumab in Children and Young Adults With INI1-Negative Cancers

Inclusion Criteria: * All participants must have one of the following histologically confirmed tumors at original diagnosis or relapse: * Stratum 1 * Malignant rhabdoid tumor (MRT) * Rhabdoid tumor of the kidney (RTK) * Epithelioid sarcoma * Chordoma (poorly differentiated or de-differentiated) * Other INI1-negative or SMARCA4-deficient malignant tumors (with PI approval) * Stratum 2 * Atypical teratoid rhabdoid tumor (ATRT) * Other INI1-negative or SMARCA4-deficient primary CNS malignant tumors (with PI approval) * All participants must have tumor assessment at original diagnosis or relapse showing the following: * Loss of INI1 confirmed by immunohistochemistry (IHC), OR * Molecular confirmation of tumor bi-allelic SMARCB1 (INI1) loss or mutation when INI1 IHC is equivocal or unavailable * Loss of SMARCA4 confirmed by IHC or molecular confirmation of tumor bi-allelic SMARCA4 loss or mutation when SMARCA4 is equivocal or unavailable * Relapsed or refractory disease and no standard treatment options as determined by locally or regionally available standards of care and treating physician's discretion * Measurable disease as defined by RECIST v1.1 (Stratum 1) or RANO criteria (Stratum 2) * Karnofsky performance status ≥ 50% for participants ≥16 years of age and Lansky performance status ≥ 50% for participants \<16 years of age * Participants must have fully recovered from the acute toxic effects of all prior anti-cancer therapy. Participants must meet the following minimum washout periods prior to first day of study treatment: * Myelosuppressive chemotherapy: At least 14 days after the last dose of myelosuppressive chemotherapy * Radiotherapy * At least 14 days after local palliative XRT (small port) * At least 90 days must have elapsed after prior TBI, craniospinal XRT or if \>50% radiation of pelvis * At least 42 days must have elapsed if other substantial BM radiation * At least 42 days must have passed since last radionuclide therapy (e.g. samarium or radium) * Small molecule biologic therapy: At least 7 days following the last dose of a nonmonoclonal biologic agent * Monoclonal antibody: At least 21 days after the last dose * Myeloid growth factors: At least 14 days following the last dose of long-acting growth factor (e.g. Neulasta) or 7 days following short-acting growth factor * Stem Cell or Autologous T Cell Infusion: At least 42 days must have elapsed after stem cell or autologous T cell infusion * Participants must have adequate organ function as defined below * Bone Marrow Function * Absolute neutrophil count ≥500/uL * Platelets ≥50,000/uL and transfusion independent * Hepatic Function * Total bilirubin ≤ 1.5 x upper limit of normal for age * ALT (SGPT) ≤ 3 x upper limit of normal * Renal function * A serum creatinine within protocol limits based on age/sex. OR * Creatinine clearance ≥ 70 mL/min/1.73 m2 for participants with creatinine levels above institutional normal * Adequate Pulmonary Function Defined as: no evidence of dyspnea at rest, no exercise intolerance due to pulmonary insufficient and a pulse oximetry \> 92% while breathing room air * Adequate pancreatic function defined as serum lipase ≤ ULN at baseline * Negative B-HCG pregnancy test in females of childbearing potential (must be drawn within 24 hours prior to initial administration of nivolumab) * Women of childbearing potential (WOCBP) receiving nivolumab agree to adhere to contraception for a period of 5 months after the last dose of nivolumab. Men receiving nivolumab and who are sexually active with WOCBP will agree to adhere to contraception for a period of 7 months after the last dose of nivolumab. * Ability to understand and/or the willingness of the patient (or parent or legally authorized representative, if minor) to provide informed consent using an institutionally approved informed consent procedure. Exclusion Criteria: * Participants who are receiving any other investigational agents. * Participants must not be receiving concomitant systemic steroid medications The use of physiologic doses of corticosteroids (up to 5 mg/m2/day prednisone equivalent) may be approved after consultation with the PI (treatment with topical, inhaled or ophthalmic corticosteroid is acceptable) * Participants with a known history of HIV, hepatitis B, and/or hepatitis C * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or any other concurrent disease which in the judgment of the Investigator would make the subject inappropriate for enrollment on this study * Has a history of (non-infectious) pneumonitis that required steroids or has current pneumonitis * Has active autoimmune disease that has required systemic treatment in the past 12 months, or a documented history of clinically severe autoimmune disease, or a syndrome that requires systemic steroids or immunosuppressive agents. Subjects with vitiligo or resolved childhood asthma/atopy are exceptions. Intermittent use of bronchodilators or local steroid injections are not excluded. Replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. Autoimmune diagnoses not listed must be approved by the Principal Investigator. * Patients who have received prior solid organ transplantation are not eligible. * Pregnancy or Breast-Feeding. Pregnant or breast-feeding women will not be entered on this study due to risks of fetal and teratogenic adverse events as there is yet no available information regarding human fetal or teratogenic toxicities. Pregnancy tests must be obtained in girls who are post-menarchal. * Has received prior therapy with an anti-PD-1, anti-PD-L1, anti-PD-L2 or anti-CTLA4 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (e.g. OX-40, CD137) * Participants who have received live / attenuated vaccine within 30 days of first dose of study treatment.