Safety and Immune Response of BMS-936559 in HIV-Infected People Taking Combination Antiretroviral Therapy

Inclusion Criteria: * HIV-1 infection, documented by any licensed rapid HIV test or HIV enzyme or chemiluminescence immunoassay (E/CIA) test kit at any time prior to study entry and confirmed by a licensed Western blot or a second antibody test by a method other than the initial rapid HIV and/or E/CIA, or by HIV-1 antigen, plasma HIV-1 RNA assay. More information on this criterion is available in the protocol. * Receiving a stable cART regimen containing at least three agents (not including ritonavir if less than a 200 mg total daily dose) with no changes in the components of their antiretroviral therapy for at least 90 days prior to study entry. NOTE: One of the agents must include an integrase inhibitor, a non-nucleoside reverse transcriptase inhibitor (NNRTI), or a boosted protease inhibitor (PI). * CD4 cell count greater than or equal to 350 cells/mm\^3 obtained within 90 days prior to study entry at any U.S. laboratory that has a clinical laboratory improvement amendments (CLIA) certification or its equivalent * Plasma HIV-1 RNA below detected limit obtained by Food and Drug Administration (FDA)-approved assays (limit of detection: 75, 50, 40, or 20) for greater than or equal to 2 years on cART. Participants must have at least one documented HIV-1 RNA less than the limit of detection 12-24 months prior to screening and one HIV-1 RNA less than the limit of detection within 12 months prior to screening. NOTE: A single unconfirmed plasma HIV-1 RNA greater than the limit of detection but less than 1,000 copies/mL is allowed if followed by HIV-1 RNA below detectable limits, but none in the 6 months prior to screening. * Plasma HIV-1 RNA level of less than 40 copies/mL obtained by the Abbott m2000 assay or less than 20 copies/mL by the Roche Taqman v2.0 assay within 90 days prior to entry. The protocol team should be notified as soon as possible if the HIV-1 RNA level is above the limit of detection for either assay. * Plasma HIV-1 RNA greater than or equal to 0.4 copies/mL by single copy assay (SCA) within 120 days prior to entry * The following laboratory values obtained within 60 days prior to entry by any U.S. laboratory that has a CLIA certification or its equivalent: * Absolute neutrophil count (ANC) greater than or equal to 1,000 cells/mm\^3 * Hemoglobin greater than or equal to 14.0 g/dL for men and greater than or equal to 12.0 g/dL for women * Platelet count greater than or equal to 75,000/mm\^3 * Creatinine clearance greater than or equal to 50 mL/min estimated by the Cockcroft-Gault equation. NOTE: A program for calculating creatinine clearance by the Cockcroft-Gault method is available on www.fstrf.org. * Alanine aminotransferase (ALT) (SGPT) less than or equal to 2.5 times upper limit of normal (ULN) * A.M. cortisol within normal limits * Fasting blood sugar within normal limits * Total bilirubin less than or equal to 1.6 x ULN. NOTE: If the participant is on an atazanavir-containing therapy then a direct bilirubin should be measured instead of the total bilirubin and must be less than or equal to 1.0 mg/dL. * The following laboratory values obtained within 90 days prior to entry by any U.S. laboratory that has a CLIA certification or its equivalent: * Thyroid stimulating hormone (TSH) and free T4 level within normal limits * Hemoglobin A1c (HgbA1c) within normal limits * Hepatitis C virus (HCV) antibody negative result within 90 days prior to study entry or, if the HCV antibody result is positive, a negative HCV RNA result prior to study entry. Participants who have received HCV treatment in the last 5 years will be excluded. * Negative hepatitis B surface antigen (HBsAg) result obtained within 90 days prior to study entry * Karnofsky performance score greater than or equal to 90 within 60 days prior to entry * Documentation of the availability of the stored pre-entry plasma specimens for HIV-1 RNA SCA determination and stored pre-entry peripheral blood mononuclear cell (PBMC) specimens for CD8 T-cell assays. Sites must receive confirmation from the processing lab via phone, e-mail, or fax, that specimens have been entered into the AIDS Clinical Trials Group's (ACTG's) Laboratory Data Management System (LDMS). * Ability and willingness of participant or legal guardian/representative to provide informed consent * Ability and willingness of participant to continue cART throughout the study * Ability to construct a fully active alternative cART regimen in the event of virologic failure on the current ART regimen * An ophthalmology exam within 180 days prior to entry and a copy of the results of the exam. NOTE: Ophthalmologic exams done to meet enrollment criteria must be performed by a licensed ophthalmologist within 180 days of the study entry visit. Results of the exam must be available for review and made part of the clinical record. Exclusion Criteria: * History of malignancy within the last 5 years or current malignancy requiring cytotoxic therapy. NOTE: A history of non-melanoma skin cancer (e.g., basal cell carcinoma or squamous cell skin cancer) is not exclusionary. * History of HIV-related opportunistic infections within the last 5 years. More information on this criterion is available in the protocol. * Current chronic, acute, or recurrent bacterial, fungal, or viral (other than HIV) infections that are serious, in the opinion of the site investigator, and required systemic therapy within 30 days prior to entry * History of or active autoimmune disorders including but not limited to inflammatory bowel diseases, scleroderma, severe psoriasis, myocarditis, uveitis, pneumonitis, systemic lupus erythematosus, rheumatoid arthritis, optic neuritis, myasthenia gravis, adrenal insufficiency, hypothyroidism and/or hyperthyroidism, autoimmune thyroiditis, or sarcoidosis. More information on this criterion is available in the protocol. * History of inflammatory disorders of the eye including uveitis (iritis, endophthalmitis, scleritis, retinitis - including viral or other infectious retinitis) and chronic or recurrent post-operative inflammation. NOTE: A history of self-limited conjunctivitis, blepharitis, or hordeolum (stye) are NOT exclusions. * Previous ocular treatment with silicone oil tamponade (for complex retinal detachment) * Intraocular surgery within 90 days prior to entry or the anticipated need for intraocular surgery during the course of the study * Intraocular laser or cryotherapy within 90 days prior to entry or the anticipated need for intraocular laser or cryotherapy during the course of the study * Evidence on eye exam of active or previous ocular inflammation or uveitis * Previous history of serious ocular trauma (e.g., penetrating trauma of the eye) * Severe cataract or other ocular abnormality that precludes adequate examination of the posterior chamber and fundus * Active infection or inflammation of the eye within 30 days prior to entry that requires systemic or topical therapy or, in the opinion of the site investigator, would complicate on-study evaluation and patient safety. NOTE: A history of self-limited allergic conjunctivitis is NOT an exclusion. More information on this criterion is available in the protocol. * Immune deficiency other than HIV * Breastfeeding * Known allergy/sensitivity or any hypersensitivity to components of BMS-936559 (anti-PD-L1) or its formulation * Active drug or alcohol use or dependence that, in the opinion of the site investigator, would interfere with adherence to study requirements * Acute or serious illness, in the opinion of the site investigator, requiring systemic treatment and/or hospitalization within 30 days prior to entry * Use of immunomodulators (e.g., interleukins, interferons, cyclosporine), HIV vaccine, systemic cytotoxic chemotherapy, or investigational therapy within 60 days prior to study entry. NOTE: Participants receiving stable physiologic glucocorticoid doses, defined as prednisone less than or equal to 10 mg/day or the equivalent, will not be excluded. Stable physiologic glucocorticoid doses should not be discontinued for the duration of the study. In addition, participants receiving inhaled or topical corticosteroids will not be excluded. * Intent to use immunomodulators (e.g., IL-2, IL-12, interferons, or tumor necrosis factor \[TNF\] modifiers) during the course of the study * Any vaccination within 30 days prior to screening SCA, pre-entry, or entry. Individuals who require vaccination will delay screening SCA until 30 days post-vaccination. Alternatively, vaccinations can occur following the screening SCA, provided they occur greater than or equal to 30 days prior to pre-entry or entry. * Current HCV antiviral therapy or participants who have received HCV treatment in the last 5 years * Positive tuberculosis (TB) purified protein derivative (PPD) skin test or interferon-gamma release assay (IGRA) at screening. NOTE: Participants with a prior positive PPD or IGRA who have not completed prophylaxis treatment will be excluded. * Women of reproductive potential. More information on this criterion is available in the protocol. * History of chronic obstructive pulmonary disease (COPD) * Type I and type II diabetes mellitus * Participants weighing less than 50 kg or greater than 200 kg. NOTE: For participants weighing between 50 kg to 52.9 kg, sites must consult with the A5326 protocol team prior to enrollment. The allowable blood volume to be drawn in an 8-week period for these participants may be less than participants weighing greater than or equal to 53 kg.