Nivolumab and/or Ipilimumab With or Without Azacitidine in Treating Patients With Myelodysplastic Syndrome

Inclusion Criteria:

* Patients with MDS (up to 20% blasts) of any risk as defined as:

  * Previously untreated
  * Previously treated with hypomethylating agent (HMA) agent; patients need to have relapsed or progressed after any number of cycles of HMA therapy; patients that do not respond to HMA therapy will also be allowed in the study; relapse or progression will be measured by International Working Group (IWG) 2006 criteria; no response will be lack of clinical benefit after at least 6 cycles of HMA therapy
* Creatinine =\< 2.0 x upper limit of normal (ULN)
* Serum bilirubin =\< 2.0 x ULN
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.0 x ULN
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2
* Females of childbearing potential must have a negative serum or urine beta human chorionic gonadotropin (beta-hCG) pregnancy test result within 24 hours prior to the first dose of treatment and must agree to use an effective contraception 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 drugs; females of non-childbearing potential are those who are postmenopausal greater than 1 year or who have had a bilateral tubal ligation or hysterectomy
* Males who have partners of childbearing potential must agree to use an effective contraceptive method during the study and a period of 31 weeks after the last dose of investigational drug
* Patients or their legally authorized representative must provide written informed consent

Exclusion Criteria:

* Second malignancy currently requiring active therapy, except breast or prostate cancer stable on or responding to endocrine therapy
* Any major surgery, radiotherapy, chemotherapy, biologic therapy, immunotherapy, experimental therapy within 2 weeks prior to the first dose of the study drugs
* Patients with any other known concurrent severe and/or uncontrolled medical condition (e.g. uncontrolled diabetes; cardiovascular disease including congestive heart failure New York Heart Association \[NYHA\] class III or IV, myocardial infarction within 6 months, and poorly controlled hypertension; chronic renal failure; or active uncontrolled infection) which, in the opinion of the investigator could compromise participation in the study
* Patients unwilling or unable to comply with the protocol
* History of pneumonitis
* Patients who are on high dose steroid (equivalent of prednisone more than 10 mg a day) or immune suppression medications
* Patients with autoimmune diseases (e.g., rheumatoid arthritis, systemic progressive sclerosis \[scleroderma\], systemic lupus erythematosus, autoimmune vasculitis \[e.g., Wegener's granulomatosis\])
* Patients with a history of inflammatory bowel disease such as Crohn's disease and ulcerative colitis
* Patients known to be positive for hepatitis B surface antigen expression or with active hepatitis C infection (positive by polymerase chain reaction or on antiviral therapy for hepatitis C within the last 6 months); patients with history of human immunodeficiency virus (HIV) disease are also excluded from the study
* Current therapy with other systemic anti-neoplastic or anti-neoplastic investigational agents
* Females who are pregnant or lactating
* Prior treatment with allogeneic stem cell transplantation