NIMBUS: Nivolumab Plus Ipilimumab in Metastatic Hypermutated HER2-negative Breast Cancer

Inclusion Criteria: * Participants must have histologically or cytologically confirmed invasive breast cancer, with metastatic disease. Participants without pathologic or cytologic confirmation of metastatic disease should have unequivocal evidence of metastasis from physical examination or radiologic evaluation. * Breast cancer must be HER2-negative by IHC or non-amplified as determined by the current ASCO-CAP criteria. If patient has more than one histological result, the most recent one will be usedfor inclusion. Participants may be ER/PR positive or negative. * Patients must harbor tumors with total mutational burden (TMB) of at least 9 mutations per megabase assessed by a cancer-gene panel containing more than 300 genes, and performed in a CLIA verified laboratory. Tests like Foundation One, Oncopanel (DFCI), or IMPACT (MSKCC) are acceptable for including patients on this trial. * Participants must have measurable disease by RECIST version 1.1. * Participants must agree to undergo a research biopsy, if tumor is safely accessible, at baseline and at day 29 cycle 1 (+14 scheduling window). Previously collected archival tissue will be obtained on all participants. Participants for whom newly-obtained samples cannot be provided (e.g. inaccessible or participant safety concern) may submit an archived specimen (block or if not possible, 20 unstained slides). * Prior chemotherapy: Participants may have received 0-3 prior chemotherapeutic regimens for metastatic breast cancer and must have been off treatment with chemotherapy for at least 14 days prior to study treatment initiation. * Patients with hormone receptor positive breast cancer must have progressed on at least one prior line of endocrine therapy in the metastatic setting or have disease recurrence while on adjuvant endocrine therapy. * Participants should also be adequately recovered from acute toxicities of prior treatment, with the exception of alopecia and peripheral sensory neuropathy. * Prior biologic therapy: Patients must have discontinued all biologic therapy at least 14 days prior to study treatment initiation. * Prior radiation therapy: Patients may have received prior radiation therapy in either the metastatic or early-stage setting. Radiation therapy must be completed 14 days prior to study treatment initiation. * In all cases, there must be no ongoing complications from prior radiotherapy. * The subject is ≥18 years old. * ECOG performance status ≤1(Karnofsky ≥70%, see Appendix A). * Participants must have normal organ and marrow function as defined below: * absolute neutrophil count ≥1,000/mcL * platelets ≥100,000/mcL * hemoglobin ≥ 8 g/dl * total bilirubin ≤1.5 × institutional upper limit of normal (ULN) (or ≤ 2.0 x ULN in patients with documented Gilbert's Syndrome) * AST(SGOT)/ALT(SGPT) ≤2.5 × institutional ULN or (≤ 3 × institutional ULN for participants with documented liver metastases) * creatinine ≤1.5 × institutional ULN OR creatinine clearance ≥ 40 mL/min (using Cockcroft-Gault formula) for participants with creatinine levels above institutional ULN. * Female subjects of childbearing potential must have a negative pregnancy test (serum or urine) at screening. * Childbearing potential is defined as: participants who have not reached a postmenopausal state (≥ 12 continuous months of amenorrhea with no identified cause other than menopause) and has not undergone surgical sterilization (removal of ovaries and/or uterus). * Female and male participants of childbearing potential must agree to use an adequate method of contraception. For women, contraception is required starting with the first dose of study medication through 150 days (5 months) after the last dose of study medication. For men who are sexuall active with women of childbearing potential, contraception is required starting with the first dose of study medication for a period of 7 months after the last dose of nivolumab. Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, established and proper use of hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception. * Participants on bisphosphonates may continue receiving bisphosphonate therapy during study treatment. Initiation of bisphosphonate or RANKL agent is allowed on study. * The participant is capable of understanding and complying with the protocol and has signed the informed consent document. Exclusion Criteria: * Major surgery within 2 weeks before the first dose of study treatment. * Concurrent administration of other anti-cancer therapy within 14 days of starting protocol therapy and during the course of this study. * The participant has received another investigational agent within 14 days of the first dose of study drug. * Has received prior therapy with an anti-PD-1, anti-PD-L1, anti-PD-L2, or anti-Cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody. * Known brain metastases that are untreated, symptomatic, or require therapy to control symptoms. Participants with a history of treated central nervous system (CNS) metastases are eligible. Treated brain metastases are defined as those having no evidence of progression for ≥ 2 weeks after treatment, and no ongoing requirement for corticosteroids, as ascertained by clinical examination and brain imaging (magnetic resonance imaging or CT scan) completed during screening. Subject must be either off corticosteroids, or on a stable or decreasing dose of ≤ 10 mg daily (or equivalent) for at least 7 days prior to first study treatment. Treatment for brain metastases may include whole brain radiotherapy, radiosurgery, or a combination as deemed appropriate by the treating physician. Participants with CNS metastases treated by neurosurgical resection or brain biopsy performed within 28 days before study treatment initiation will be excluded. * The subject has uncontrolled, significant intercurrent or recent illness. Individuals with a history of different malignancy are ineligible except for the following circumstances. Individuals with a history of other malignancies are eligible if they have been disease-free for at least 3 years or are deemed by the investigator to be at low risk for recurrence of that malignancy. * Participant has an active infection requiring IV antibiotics at initiation of study therapy. * Patient has a medical condition that requires chronic systemic steroid therapy or on any other form of immunosuppressive medication. For example, participants with autoimmune disease that requires systemic steroids or immunosuppression agents should be excluded. Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Subjects with current pneumonitis, or requiring supplementary O2 therapy. * The participant is known to be positive for human immunodeficiency virus (HIV), HepBsAg, or HCV RNA. HIV-positive participants on combination antiretroviral therapy are ineligible * Participants with any other active malignancy requiring concurrent intervention. * Known hypersensitivity to any of the components of ipilimumab or nivolumab. * The participant has received a live vaccine within 28 days prior to the first dose of trial treatment and while participating in the trial. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster, yellow fever, rabies, BCG, and typhoid vaccine. The use of the inactivated seasonal influenza vaccine (Fluzone®) is allowed. * The participant is pregnant or breastfeeding.