This Phase 1 trial will investigate the safety of a modified stem cell transplant procedure for treating advanced breast cancer. Patients with cancers can sometimes benefit greatly from transplants of stem cells (cells produced by the bone marrow that mature into blood cells). In addition to producing new bone marrow and restoring normal blood production and immunity, the donated cells fight any residual tumor cells that might have remained in the body, in what is called a "graft-versus-tumor" effect. However, severe problems, or sometimes even death, may follow these transplants as a result of the high-dose chemotherapy and radiation that accompany the procedure. Also, donated immune cells called lymphocytes, or T cells, sometimes attack healthy tissues in a reaction called graft-versus-host-disease (GVHD), damaging organs such as the liver, intestines and skin. This study will use the following strategies to try to reduce these risks: