T-Cell Infusion, Aldesleukin, and Utomilumab in Treating Patients With Recurrent Ovarian Cancer

Inclusion Criteria: * Histopathologic documentation (must be performed or reviewed at MD Anderson) of recurrent high grade epithelial ovarian cancer. * At least one prior line of platinum-based chemotherapy (subjects are eligible for enrollment and leukapheresis while still platinum-sensitive, however, they must have developed platinum resistant disease for treatment (turnstile 2). * Tumor expressing PRAME and/or COL6A3. * Expression of HLA-A\*0201. * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1. * Expected survival of greater than 16 weeks. * Willing and able to give informed consent. * Hemoglobin \>= 9.0 g/dL. * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9/L (\>=1000 per mm\^3). * Platelet count \>= 75 x 10\^9/L (\>=100,000 per mm\^3). * Serum bilirubin =\< 1.5 x institutional upper limit of normal (ULN) unless diagnosed with Gilbert's syndrome. * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN unless liver metastases are present, in which case it must be =\< 5x ULN. * Serum creatinine clearance (CL) \>40 mL/min by the Cockcroft-Gault formula or by 24-hour urine collection for determination of creatinine clearance. * Subjects must either be of non-reproductive potential (i.e., post-menopausal by history: \>= 50 years old and no menses for \>=1 year without an alternative medical cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry. * 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. * {Turnstile 2} Subjects must have platinum resistant disease (progression on a platinum-containing regimen or recurrence within 180 days of last dose of platinum-containing chemotherapy). Subjects that are not platinum resistant but are deemed not to be candidates for platinum-based chemotherapy due to prior significant allergic reaction may participate with principal investigator (PI) approval. * {Turnstile 2} Bi-dimensionally measurable disease by radiographic imaging (magnetic resonance imaging \[MRI\] or computed tomography \[CT\] scan). * {Turnstile 2} At least 4 weeks must have elapsed since the last chemotherapy, immunotherapy, radiotherapy or major surgery. At least 6 weeks for bevacizumab. * {Turnstile 2} Toxicity related to prior therapy must either have returned to =\< grade 1, baseline, or been deemed irreversible. Exclusion Criteria: * 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) \< 60% of normal or carbon monoxide diffusing capacity (DLco) (corrected \[corr\] for hemoglobin \[Hgb\]) \< 55% will be excluded. * Significant cardiovascular abnormalities including any one of the following: Congestive heart failure, Clinically significant hypotension, symptoms of coronary artery disease, presence of cardiac arrhythmias on electrocardiography (EKG) requiring drug therapy; or patients with a history of cardiovascular disease. (Patients with the above will undergo a cardiac evaluation which can include a stress test and/or echocardiography. Results of this evaluation will be considered before excluding patients on the basis of cardiovascular abnormalities). Subjects with evidence of stress-induced cardiac ischemia or ejection fraction less than 55% will be excluded. * History of central nervous system (CNS) metastasis. * 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. * {Turnstile 2} Participation in another clinical study with an investigational product administered during the last 28 days. * {Turnstile 2} Receipt of the last dose of previous chemotherapy, hormonal, or biologic treatment for ovarian, fallopian tube, or primary peritoneal cancer in the last 28 days (in the last 6 weeks for bevacizumab). * {Turnstile 2} Current or prior use of immunosuppressive medication within 28 days before enrollment, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid. * {Turnstile 2} Any prior grade \>= 3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent, or any unresolved irAE \> grade 1. * History of allogeneic organ transplant. * Unresolved partial or complete small or large bowel obstruction. * {Turnstile 2} Receipt of live attenuated vaccination within 30 days prior to enrollment or within 30 days of planned lymphodepletion. * 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. * Active viral hepatitis. * Confirmed human immunodeficiency virus (HIV) infection.