Ipilimumab or Nivolumab in Treating Patients With Relapsed Hematologic Malignancies After Donor Stem Cell Transplant

Inclusion Criteria:

* Histologically or cytologically confirmed hematologic malignancy
* The following malignancies will be considered eligible if progressive or persistent:

  * Chronic lymphocytic leukemia (CLL)
  * Non-Hodgkin lymphoma (NHL)
  * Hodgkin lymphoma (HL)
  * Multiple myeloma (MM)
  * Acute leukemia (acute myeloid leukemia \[AML\], acute lymphoblastic leukemia \[ALL\])
  * Myelodysplastic syndrome (MDS)
  * Myeloproliferative neoplasms (MPN)
  * Chronic myeloid leukemia (CML)
* Life expectancy of greater than 3 months
* Must have undergone allogeneic hematopoietic stem cell transplantation (HSCT) (regardless of stem cell source)
* Must have baseline donor T cell chimerism of \>= 20%
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2
* Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (unless due to Gilbert's disease or disease-related hemolysis, then =\< 3.0 x ULN)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional ULN
* Creatinine =\< 1.5 x institutional ULN
* Prednisone dose =\< 5 mg/day and off all other systemic immunosuppressive medications for at least 4 weeks prior to study entry
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Participants who have had anti-tumor therapy or other investigational agents within 4 weeks prior to registration (6 weeks for nitrosoureas or mitomycin C), or those who have not recovered from adverse events due to agents administered more than 4 weeks prior to registration
* Patients with prior history of or active severe (grade 3 or 4) acute graft-versus-host disease (GVHD)
* Patients with a history of prior treatment with ipilimumab, anti-programmed cell death protein 1 (PD 1) antibody, or cluster of differentiation (CD)137 agonist therapy are ineligible for the ipilimumab arm, but are eligible for the nivolumab arm
* Patients who have had donor lymphocyte infusion (DLI) within 8 weeks prior to registration
* Autoimmune disease: Patients with a history of inflammatory bowel disease, including ulcerative colitis and Crohn's disease, are excluded from this study, as are patients with a history of symptomatic disease (e.g., rheumatoid arthritis, systemic progressive sclerosis \[scleroderma\], systemic lupus erythematosus, autoimmune vasculitis \[e.g., Wegener's granulomatosis\]) and motor neuropathy considered of autoimmune origin (e.g. Guillain-Barre syndrome and myasthenia gravis); patients with Hashimoto's thyroiditis are eligible to go on study
* Patients with known chronic human immunodeficiency virus (HIV), hepatitis B or hepatitis C infections should be excluded
* 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
* Pregnant women are excluded from this study because ipilimumab and nivolumab are immunomodulatory agents with the potential for teratogenic or abortifacient effects; the effects of ipilimumab and nivolumab on the developing human fetus are unknown; for this reason and because immunomodulatory agents are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study or within 23 weeks after the last dose of study drug, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and for at least 31 weeks after completion of ipilimumab or nivolumab administration