This study will evaluate the safety and effectiveness of stem cell transplantation in which the donor's T cells (a type of lymphocyte, or white blood cell) are removed and then added back. Certain patients with bone marrow malignancies undergo transplantation of donated stem cells to generate new and normally functioning bone marrow. However, T-cells from the donor may see the patient's cells as foreign and mount an immune response to reject them, causing what is called "graft-versus-host-disease" (GVHD). Therefore, in this protocol, T-cells are removed from the donor cells to prevent this complication. However, because T-cells are important in fighting viral infections as well as any remaining malignant cells (called graft-versus-leukemia effect), the donor T-cells are given to the patient (added back) at a later time after the transplant when they can provide needed immunity with less risk of causing GVHD.