Monoclonal Antibody Therapy in Treating Patients With Ovarian Epithelial Cancer, Melanoma, Acute Myeloid Leukemia, Myelodysplastic Syndrome, or Non-Small Cell Lung Cancer

Inclusion Criteria: * Patients previously vaccinated with GM-CSF-based vaccines using lethally irradiated, autologous melanoma, ovarian cancer, acute myelogenous leukemia/myelodysplasia, or non-small cell lung cancer cells; patients with acute myelogenous leukemia/myelodysplasia or non-small cell lung cancer who have not been vaccinated with an autologous, GM-CSF based vaccine * \>= 4 weeks since treatment (chemo-, radiation, hormone, immuno-, etc., therapy) * Patients must have recovered from any acute toxicity associated with prior therapy * Measurable epithelial ovarian cancer, melanoma, AML/MDS, or non-small cell lung cancer * No standard curative treatment options * Not require immediate palliative therapy * Patients with epithelial ovarian cancer must have persistent or recurrent disease following primary surgery and primary chemotherapy * Patients with melanoma must be stage IV disease * Patients with AML/MDS, but without MDS, must be: a) in second relapse or b) first relapse with no option for bone marrow transplant or c) not a candidate for immunosuppressive chemotherapy due to age or comorbid disease * Patients with non-small cell lung cancer must be not curable by standard surgery, chemotherapy, and/or radiation * Life expectancy \>= 12 weeks * ECOG performance status of 0, 1 or 2 * Written informed consent * Due to the unknown effects of MDX-CTLA-4 on the fetus or nursing infant, pregnant or nursing women should not be included; women should be either: post-menopausal for at least 1 year; surgically incapable of bearing children; or utilizing an intrauterine device, and/or spermicide and barrier, for contraception; during the study, use of oral contraception alone is not acceptable; women of childbearing potential must have a negative serum beta-HCG pregnancy test conducted during screening, and a negative urinary beta-HCG pregnancy test conducted within 24 hours prior to treatment; due to the unknown effects of MDX-CTLA-4 on the fetus, men should not father children during the study * WBC \> 1,000 cells/mm\^3 (except for AML/MDS patients) * Serum creatinine \< 2 mg/dL * Platelets \> 75,000 cells/mm\^3 (except for AML/MDS patients) * AST and ALT \< 2 x UNL * Total bilirubin \< 2 x UNL Exclusion Criteria: * Active infection * Autoimmune disease requiring immunosuppressive treatment * Any underlying medical condition which, in the principal investigator's opinion, will make the administration of study drug hazardous or obscure the interpretation of adverse events * Any concurrent medical condition requiring the use of systemic steroids (use of inhaled or topical steroids is acceptable) * CNS metastases, unless previously treated and stable for at least three months * Patients who have received prior treatment with MDX-CTLA-4