Laboratory-Treated T Cells and Ipilimumab in Treating Patients With Metastatic Melanoma

Inclusion Criteria: * Histopathologic documentation of melanoma concurrent with the diagnosis of metastatic disease * Expression of human leukocyte antigen (HLA)-A2 * Eastern Cooperative Oncology Group (ECOG)/Zubrod performance status of 0-1 * Women of childbearing potential (WOCBP) must be using an adequate method of contraception to avoid pregnancy throughout the study in such a manner that the risk of pregnancy is minimized; suggested precautions should be used to minimize the risk or pregnancy for at least 1 month before start of therapy, and while women are on study for up to 3 months after T cell infusion, and at least 8 weeks after the study drug is stopped; WOCBP include any female who has experienced menarche and who has not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation or bilateral oophorectomy) or is not postmenopausal * Men must be willing and able to use an acceptable method of birth control, for at least 3 months after completion of the study, if their sexual partners are WOCBP * Willing and able to give informed consent * Adequate venous access-consider peripherally inserted central catheter (PICC) or central line * Bi-dimensionally measurable disease by palpation on clinical exam, or radiographic imaging (X-ray, computed tomography \[CT\] scan) * At least 4 weeks must have elapsed since the last chemotherapy, radiotherapy or major surgery; at least 6 weeks for nitrosoureas, mitomycin C and liposomal doxorubicin; if started before T-cell administration, Ipilimumab infusions must be least 21 days apart * Toxicity related to prior therapy must either have returned to =\< grade 1, baseline, or been deemed irreversible * Persons of reproductive potential must agree to use and utilize an adequate method of contraception throughout treatment and for at least 8 weeks after study drug is stopped Exclusion Criteria: * Patients with active infections or oral temperature \> 38.2 C within 72 hours prior to planned leukapheresis; the procedure may be deferred * Patients with hematocrit (Hct) \< 30%, white blood cells (WBC) \< 2500/uL and platelets \< 50,000 immediately prior to leukapheresis; the procedure may be deferred * Any other malignancy from which the patient has been disease-free for less than 5 years, with the exception of adequately treated and cured basal or squamous cell skin cancer, superficial bladder cancer, carcinoma in situ of the cervix * White blood cell count (WBC) \< 2000/uL * Hematocrit (Hct) \< 24% or hemoglobin (Hb) \< 8 g/dL * Absolute neutrophile count (ANC) \< 1000 * Platelets \< 50,000 * Creatinine \> 3.0 x upper limit normal (ULN) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) \> 2.5 x ULN * Bilirubin \> 3 x ULN * Pregnant women, nursing mothers, men or women of reproductive ability who are unwilling to use effective contraception; women of childbearing potential with a positive pregnancy test within 3 days prior to entry * Clinically significant pulmonary dysfunction, as determined by medical history and physical exam; patients so identified will undergo pulmonary functions testing and those with forced expiratory volume in one second (FEV1) \< 2.0 L or diffusion capacity of carbon monoxide (DLco) (corr for Hgb) \< 50% will be excluded * Significant cardiovascular abnormalities as defined by any one of the following: * Congestive heart failure * Clinically significant hypotension * Symptoms of coronary artery disease * Presence of cardiac arrhythmias on electrocardiogram (EKG) requiring drug therapy * Ejection fraction \< 50 % (echocardiogram or multi gated acquisition \[MUGA\] scan) * Active and untreated central nervous system (CNS) metastasis (including metastasis identified during screening magnetic resonance imaging \[MRI\] or contrast CT) * Autoimmune disease: Patients with a history of Inflammatory Bowel Disease are excluded from this study, as are patients with a history of autoimmune disease (e.g. systemic lupus erythematosus, vasculitis, infiltrating lung disease) whose possible progression during treatment would be considered by the Investigator to be unacceptable * Any underlying medical or psychiatric condition, which in the opinion of the Investigator, will make the administration of study drug hazardous or obscure the interpretation of adverse events, such as a condition associated with frequent diarrhea * Positive screening tests for human immunodeficiency virus (HIV), hepatitis (Hep) B, and Hep C; if positive results are not indicative of true active or chronic infection, the patient can be treated * Steroids are not permitted 3 days prior to T cell infusion and concurrently during therapy * No prisoners or children will be enrolled on this study * Any non-oncology vaccine therapy used for the prevention of infectious disease within 1 month before or after any ipilimumab dose * Patients may not be on any other treatments for their cancer aside from those included in the protocol; patients may not undergo another form of treatment concurrently with this study