Nivolumab and Ipilimumab in Treating Patients With Metastatic Uveal Melanoma

Inclusion Criteria: * Willing and able to give written informed consent * History of uveal melanoma and documented metastatic disease with at least one measurable lesion is required; which is \>= 1 cm x 1 cm (on spiral computed tomography \[CT\] or equivalent) * Any number of prior therapies is allowed * White blood cell (WBC) \>= 2000/uL * Absolute neutrophil count (ANC) \>= 1500/uL * Platelets \>= 100 x 10\^3/uL * Hemoglobin \>= 9 g/dL * Creatinine =\< 1.5 x upper limit of normal (ULN) or creatinine clearance (CrCl) \> 40 mL/min (using the Cockcroft-Gault formula) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 3 x ULN for patients without liver metastasis, =\< 5 x ULN for liver metastases * Bilirubin =\< 1.5 x ULN, (except patients with Gilbert's syndrome, who must have a total bilirubin less than 3.0 mg/dL) * In suspected patients no active or chronic infection with human immunodeficiency virus (HIV), hepatitis B, or hepatitis C * Performance status Eastern Cooperative Oncology Group (ECOG) 0-1 * Baseline imaging in the form of CT chest, abdomen, pelvis with oral and intravenous contrast within 28 days of study entry; for patients with a contrast allergy, choice of alternative body imaging will be at the discretion of the investigator or his designee; magnetic resonance imaging (MRI) of the brain is only needed if clinically indicated * Prior to start of treatment must be more than 21 days elapsed from surgery, radiation therapy, or prior chemotherapy; more than 42 days elapsed from prior immune therapy including vaccines * Women of childbearing potential (WOCBP) and fertile men with partners of childbearing potential must be using an adequate method of contraception to avoid pregnancy throughout the study and for up to 26 weeks after the last dose of investigational product, in such a manner that the risk of pregnancy is minimized Exclusion Criteria: * Untreated primary uveal melanoma except in cases where metastatic disease is diagnosed at the time of primary disease * Metastatic uveal melanoma patients with bone-only disease * Any other malignancy from which the patient has been disease-free for less than 2 years, with the exception of adequately treated and cured basal or squamous cell skin cancer, superficial bladder cancer or carcinoma in situ of the cervix, breast, or prostate * 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 \[e.g., Wegener's Granulomatosis\]; motor neuropathy considered of autoimmune origin (e.g. Guillain-Barre Syndrome and Myasthenia Gravis) * Any underlying medical or psychiatric condition, which in the opinion of the investigator will make the administration of ipilimumab hazardous or obscure the interpretation of adverse events (AEs), such as a condition associated with frequent diarrhea * Any non-oncology vaccine therapy used for prevention of infectious diseases (for up to 1 month before or after any dose of ipilimumab) * Concomitant therapy with any of the following: tamoxifen, toremifene, IL 2, interferon, or other non-study immunotherapy regimens; cytotoxic chemotherapy; immunosuppressive agents; other investigation therapies; or chronic use of systemic corticosteroids greater than physiologic replacement doses; ocular steroid use is acceptable; (a) concomitant palliative radiation for the purposes of symptom management is allowed * Women of childbearing potential (WOCBP) who: (a) are unwilling or unable to use an acceptable method of contraception to avoid pregnancy for their entire study period and for up to 26 weeks after cessation of study drug, or (b) have a positive pregnancy test at baseline, or (c) are pregnant or breastfeeding * Prisoners or subjects who are compulsorily detained (involuntarily incarcerated) for treatment of either a psychiatric or physical (e.g., infectious) illness