Nivolumab With or Without Ipilimumab in Treating Patients With Metastatic Sarcoma That Cannot Be Removed by Surgery

Inclusion Criteria: * PRE-REGISTRATION ELIGIBILITY CRITERIA: * Patients must have a formalin-fixed, paraffin-embedded (FFPE) tumor block OR 1 representative hematoxylin and eosin (H\&E) and 20 unstained sarcoma tissue slides available for submission to central pathology review; this review is mandatory prior to registration to confirm eligibility * REGISTRATION ELIGIBILITY CRITERIA: * Patients must have histologically confirmed bone or soft tissue sarcoma by central pathology review * Patients must have histologically confirmed liposarcoma (LPS) (only dedifferentiated and pleomorphic; well differentiated not eligible), undifferentiated pleomorphic sarcoma (UPS)/malignant fibrous histiocytoma (MFH), or gastrointestinal stromal tumor (GIST) * Measurable disease * Locally advanced/unresectable or metastatic disease * \>= 1 prior systemic therapy for sarcoma, including adjuvant systemic therapy * No prior therapy with ipilimumab or nivolumab, or any agent targeting programmed cell death 1 (PD-1), PD-L1 or cytotoxic T-lymphocyte-associated protein 4 (CTLA-4) * No treatment with biologic therapy, immunotherapy, chemotherapy, investigational agent for malignancy, or radiation =\< 28 days before study registration; no treatment with nitrosourea or mitomycin =\< 42 days before study registration; for GIST, tyrosine kinase inhibitor can be continued for up to 3 days prior to initiation of study treatment * Patients should have resolution of any toxic effects of prior therapy (except alopecia) to NCI CTCAE, version 4.0, grade 1 or less * No history of the following: * Active known or suspected autoimmune disease * Patients with human immunodeficiency virus (HIV) are eligible if the lymphocytes \> 350 cluster of differentiation (CD)4+ cells and no detectable viral load * Symptomatic, untreated, or uncontrolled brain metastases present * Active autoimmune colitis * Autoimmune panhypopituitarism * Autoimmune adrenal insufficiency * Known active hepatitis B or C * Hepatitis B can be defined as: * Hepatitis B surface antigen (HBsAg) \> 6 months * Serum hepatitis B virus (HBV) deoxyribonucleic acid (DNA) 20,000 IU/ml (105 copies/ml), lower values 2,000-20,000 IU/ml (104-105 copies/ml) are often seen in hepatitis B e antigen (HBeAg)-negative chronic hepatitis B * Persistent or intermittent elevation in alanine aminotransferase (ALT)/alanine aminotransferase (AST) levels * Liver biopsy showing chronic hepatitis with moderate or severe necroinflammation * Hepatitis C can be defined as: * Hepatitis C antibody (Ab) positive * Presence of hepatitis C virus (HCV) ribonucleic acid (RNA) * Known active pulmonary disease with hypoxia defined as: * Oxygen saturation \< 85% on room air or * Oxygen saturation \< 88% despite supplemental oxygen * No systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration * Not pregnant and not nursing because this study involves an agent that has known genotoxic, mutagenic and teratogenic effects; therefore for women of childbearing potential only, a negative pregnancy test done =\< 7 days prior to registration is required * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Creatinine =\< 1.5 x upper limit of normal (ULN) OR calculated (calc.) creatinine clearance \> 45 mL/min using the lean body mass formula only (Modified Cockcroft and Gault; Shargel and Yu 1985) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) in absence of Gilbert disease (total bilirubin =\< 3 x ULN with Gilbert); also, if hyperbilirubinemia is clearly attributed to liver metastases total bilirubin =\< 3 x ULN is permitted * AST/ALT =\< 3 x upper limit of normal (ULN) * Thyroid stimulating hormone (TSH) within normal limits (WNL); supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Measurable disease * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Locally advanced/unresectable or metastatic disease * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Patient MUST have had progressive disease (radiographic or clinical) while on arm 1 single agent nivolumab while registered to A091401 * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Patients removed from any immunotherapy for reasons other than progressive disease, including arm 1 single agent nivolumab of A091401, are NOT eligible for re-registration * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Patients must have completed a minimum of 10 weeks of single agent nivolumab on arm 1 of A091401 to be eligible for re-registration * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Patients must have completed study drug on arm 1 of A091401 (i.e., last dose of nivolumab) =\< 12 months of re-registration to crossover dual agent therapy * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): No treatment with immunotherapy =\< 21 days before re-registration; no treatment with biologic therapy, chemotherapy, investigational agent for malignancy, or radiation =\< 28 days before re-registration; no treatment with nitrosourea or mitomycin =\< 42 days before re-registration * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Patients should have resolution of any toxic effects of prior therapy (except fatigue and alopecia) to NCI CTCAE, version 4.0, grade 1 or less, including immune toxicity * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): No systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of re-registration * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Not pregnant and not nursing because this study involves an agent that has known genotoxic, mutagenic and teratogenic effects; therefore, for women of childbearing potential only, a negative pregnancy test done =\< 7 days prior to re-registration is required * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): ECOG performance status 0 or 1 * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): ANC \>= 1,500/mm\^3 * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Platelet count \>= 100,000/mm\^3 * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Creatinine =\< 1.5 ULN OR calc. creatinine clearance \> 45 mL/min (using lean body mass formula only \[Modified Cockcroft and Gault; Shargel and Yu 1985\]) * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): Total bilirubin =\< 1.5 x ULN in absence of Gilbert disease (total bilirubin =\< 3 x ULN with Gilbert); if hyperbilirubinemia is clearly attributed to liver metastases, total bilirubin =\< 3 x ULN is permitted * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): AST/ALT =\< 3 x ULN * RE-REGISTRATION ELIGIBILITY CRITERIA (FOR PATIENTS WHO CROSSOVER FROM ARM 1 NIVOLUMAB ALONE TO DUAL AGENT NIVOLUMAB AND IPILIMUMAB UPON PROGRESSION): TSH WNL; supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH, if free T4 is normal and patient is clinically euthyroid, patient is eligible