Phase II Randomized Trial of Nivolumab With or Without Ipilimumab in Patients With Persistent or Recurrent Epithelial Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

• Patients must have recurrent or persistent epithelial ovarian, fallopian tube, or primary peritoneal cancer with documented disease progression (disease not amendable to curative therapy); histologic confirmation of the original primary tumor is required via the pathology report; NOTE: patients with mucinous histology are NOT eligible; patients with carcinosarcoma histology are NOT eligible

• All patients must have measurable disease as defined by RECIST 1.1; measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded); each lesion must be \>= 10 mm when measured by computed tomography (CT), magnetic resonance imaging (MRI) or caliper measurement by clinical exam; or \>= 20 mm when measured by chest x-ray; lymph nodes must be \>= 15 mm in short axis when measured by CT or MRI

• Patients must have at least one target lesion to be used to assess response on this protocol as defined by RECIST 1.1; tumors within a previously irradiated field will be designated as non-target lesions unless progression is documented or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy

• Appropriate for study entry based on the following diagnostic workup:

‣ History/physical examination within 28 days prior to registration

⁃ Imaging of target lesion(s) within 28 days prior to registration

⁃ Further protocol-specific assessments:

• Recovery from effects of recent surgery, radiotherapy or chemotherapy

∙ Free of active infection requiring antibiotics (with the exception of uncomplicated urinary tract infection \[UTI\])

∙ Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration

∙ Any other prior therapy directed at the malignant tumor including chemotherapy, targeted agents, immunologic agents, and any investigational agents, must be discontinued at least 4 weeks prior to registration (6 weeks for nitrosoureas or mitomycin C)

∙ Any prior radiation therapy must be completed at least 4 weeks prior to registration

∙ At least 4 weeks must have elapsed since major surgery

• Patients are allowed to have received up to three prior cytotoxic regimens for treatment of their epithelial ovarian, fallopian tube, or primary peritoneal cancer; they must have had one prior platinum-based chemotherapeutic regimen for management of primary disease, possibly including intra-peritoneal therapy, consolidation, biologic/targeted (non-cytotoxic) agents or extended therapy (maintenance/consolidation) administered after surgical or non-surgical assessment; patients are allowed to have received, but are not required to have received, one to two cytotoxic regimens for management of recurrent or persistent disease; (for the purposes of this study poly adenosine diphosphate \[ADP\] ribose polymerase \[PARP\] inhibitors given for recurrent or progressive disease will be considered cytotoxic; PARP inhibitors given as maintenance therapy in continuation with management of primary disease will not be considered as a separate cytotoxic regimen); if two cytotoxic regimens had been received for management of recurrent or persistent disease, one of these regimens would have had to contain either a platinum or a taxane agent

• Performance status of 0, 1 or 2 within 28 days prior to registration

• Absolute neutrophil count (ANC) \>= 1,500/ul (within 14 days prior to registration)

• Platelets \>= 100,000/ul (within 14 days prior to registration)

• Creatinine =\< 1.5 x institutional/laboratory upper limit of normal (ULN) (within 14 days prior to registration)

• Bilirubin =\< 1.5 x ULN; for patients with Gilbert's syndrome, bilirubin =\< 3.0 mg/dL is acceptable (within 14 days prior to registration)

• Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN (within 14 days prior to registration)

• Albumin \>= 2.8 g/dL (within 14 days prior to registration)

• Adequate thyroid function within 28 days prior to registration defined as serum thyroid-stimulating hormone (TSH) in normal range

• The patient or a legally authorized representative must provide study-specific informed consent prior to study entry

• Platinum-free interval (PFI) - patients must have progressed \< 12 months after completion of their last platinum-based chemotherapy; the date (platinum free interval) should be calculated from the last administered dose of platinum therapy to documentation of progression

• Adequate oxygen saturation via pulse oximeter within 28 days prior to registration (i.e., patient can NOT have CTCAE hypoxia grade 2 or greater)

• Left ventricular ejection fraction (LVEF) \>= 50% (measured within 28 days of study entry)