This study will determine the recommend dose of palbociclib in combination with letrozole and another medication, Ado-trastuzumab emtansine (T-DM1). Additionally, researchers will determine how well this recommended dose will improve outcomes in this type of advanced breast cancer. The study will include a safety lead-in with escalating dosing of palbociclib to determine the recommended phase II dose (RP2D) of palbociclib in this combination and an expanded phase II of palbociclib at the RP2D in combination with letrozole and Ado- trastuzumab Emtansine (T-DM1). The starting dose of palbociclib will be 75 milligrams (mg) by mouth (PO) daily for each 21 day cycle. If 0 of 3 patients at the 75mg dose level experience a dose limiting toxicity (DLT), the next 3 patients will be enrolled at the next higher dosing cohort of 100mg PO daily for each 21 day cycle. If 0 of 3 patients at the 100mg dose level experience a DLT, the next 3 patients will be enrolled at the next higher dosing cohort of 125mg PO daily for each 21 day cycle. If 0 of 3 patients at the 125mg dose level experience a DLT, 125mg PO daily of palbociclib will be the phase II recommended dose used in the phase II expanded cohort. Patients receiving the phase II recommended dose in phase I will be enrolled in phase II of the study. During safety lead-in and expanded phase II, Letrozole 2.5mg PO will be administered daily for each 21 day cycle and T-DM1 3.6 milligrams per kilograms intravenously (IV) will be administered on Day 1 of each 21 day cycle.
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Inclusion Criteria: * Pathologically confirmed diagnosis of Estrogen Receptor (ER) positive and HER2 (human epidermal growth factor receptor 2) positive metastatic breast cancer based on local laboratory results. * Prior treatment with a taxane (including paclitaxel, docetaxel and/or nanoparticle protein-bound paclitaxel). * Prior treatment with trastuzumab with or without pertuzumab. * Measurable or non-measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) criteria version 1.1. * Eastern Cooperative Oncology Group Performance Status of 0-2 * Adequate organ and marrow function * Women must be post-menopausal * Must be able to swallow pills Exclusion Criteria: * Current or anticipated use of other investigational agents * Prior therapy with a cyclin-dependent kinase 4/6 inhibitor * Subject has received chemotherapy or radiotherapy within 14 days prior to Cycle 1, Day 1 of the study or has not recovered from adverse events due to agents administered more than 14 days earlier * Subject has leptomeningeal disease * History of allergic reactions attributed to compounds of similar chemical or biologic composition to palbociclib or other agents used in study * Subject has other illness or disease that the investigator believes will interfere with study requirements.