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