Sipuleucel-T With Immediate vs. Delayed Cytotoxic T-Lymphocyte-Associated Protein 4 (CTLA-4) Blockade for Prostate Cancer

Inclusion Criteria: 1. Histologically confirmed, metastatic prostate adenocarcinoma (positive bone scan and/or measurable disease on CT scan and/or MRI of the abdomen and pelvis). 2. Progressive disease after androgen deprivation, as defined by Prostate Cancer Clinical Trials Working Group 2 (PCWG2) and/or RECIST criteria. Patients must have disease progression by one or both of the following: * For patients with measurable disease, progression is defined as at least a 20% increase in the sum of the longest diameter (LD) of target lesions or the appearance of one or more new lesions, as per RECIST criteria version 1.1. * For patients with no measurable disease, a positive bone scan and elevated prostate specific antigen (PSA) will be required. PSA evidence for progressive prostate cancer consists of a PSA level of at least 2 ng/milliliter (mL), which has risen on at least 2 successive occasions, at least 1 week apart. If the confirmatory PSA value is not greater than the screening PSA value, then an additional test for rising PSA will be required to document progression. * If no prior orchiectomy has been performed, patients must remain on luteinizing hormone-releasing hormone (LHRH) agonist or antagonist (e.g. degarelix) therapy. Patients who are receiving an antiandrogen as part of primary androgen ablation must demonstrate disease progression following discontinuation of the antiandrogen, defined as two consecutive rising PSA values, obtained at least two weeks apart, or documented osseous or soft tissue progression. At least one of the PSA values must be obtained at least four weeks (flutamide) or six weeks (bicalutamide or nilutamide) after discontinuation. 3. Laboratory requirements: * Absolute neutrophil count (ANC) ≥ 1500/μL * Bilirubin \< 1.5 x upper limit of normal (ULN) * Hemoglobin ≥ 8 g/dL * PSA ≥ 2 ng/mL * Platelets ≥ 100,000/μL * aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 2.5 x ULN * Creatinine clearance ≥ 60 mL/min by the Cockcroft Gault equation * Testosterone less than or equal to 50 ng/dL 4. Eastern Cooperative Oncology Group (ECOG) performance status 0 - 1 and life expectancy ≥ 12 weeks. 5. At least 18 years of age or older. 6. Patients receiving any other hormonal therapy, including any dose of megestrol acetate (Megace), Proscar (finasteride), any herbal product known to decrease PSA levels (e.g. Saw Palmetto, PC-SPES), or any systemic corticosteroid, must discontinue the agent for at least four weeks prior to study treatment. Progressive disease as defined above must be documented after discontinuation of any hormonal therapy (with the exception of a LHRH agonist). 7. Prior radiation therapy must be completed ≥ 4 weeks prior to enrollment and the patient must have recovered from all toxicity. Prior radiopharmaceuticals (strontium, samarium) must be completed ≥ 8 weeks prior to enrollment. 8. Because of the unknown potential risk to a gamete and/or developing embryo from these investigational therapies, patients must agree to use adequate contraception (barrier method for males) for the duration of study participation, and for three months after discontinuing therapy. Exclusion Criteria: 1. Prior chemotherapy for prostate cancer, with the exception of neoadjuvant chemotherapy, because of the potential effect of chemotherapy on the immune system. 2. Prior sipuleucel-T treatment or investigational immunotherapy. 3. Prostate cancer pain requiring regularly scheduled narcotics. 4. Current treatment with systemic steroid therapy (inhaled/topical steroids are acceptable). Systemic corticosteroids must be discontinued for at least 4 weeks prior to first treatment. 5. History of autoimmune disease including, but not limited to: * Systemic lupus erythematosis (SLE), scleroderma, CREST syndrome, rheumatoid arthritis * Inflammatory bowel disease, celiac disease, primary biliary cirrhosis, autoimmune hepatitis * Dermatomyositis, polymyositis, giant cell arteritis * Autoimmune hemolytic anemia (AIHA), cryoglobulinemia, antiphospholipid antibody syndrome (APLS) * Diabetes mellitus type I, myasthenia gravis, Grave's disease * Wegener's granulomatosis or other vasculitis * A history of Hashimoto's thyroiditis, psoriasis, or eczema, any of which has been inactive for at least one year, or isolated Raynaud's phenomenon is acceptable 6. Known central nervous system or visceral metastases. 7. Medical or psychiatric illness that would, in the opinion of the investigator, preclude participation in the study or the ability of patients to provide informed consent for themselves. 8. Cardiovascular disease that meets one of the following: congestive heart failure (New York Heart Association Class III or IV), active angina pectoris, or recent myocardial infarction (within the last 6 months). 9. Concurrent or prior malignancy except for the following: * Adequately treated basal or squamous cell skin cancer * Adequately treated stage I or II cancer from which the patient is currently in complete remission * Any other cancer from which the patient has been disease-free for 5 years 10. Known HIV or other history of immunodeficiency disorder. 11. Prisoners or subjects who are compulsorily detained (involuntarily incarcerated) for treatment of either a psychiatric or medical (e.g. infectious) illness. 12. 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 (AE), such as a condition associated with frequent diarrhea. 13. A history of prior treatment with ipilimumab or prior cluster of differentiation 137 (CD137) agonist or CTLA-4 inhibitor or agonist.