Allogeneic Hematopoietic Stem Cell Transplantation for Severe Aplastic Anemia and Other Bone Marrow Failure Syndromes Using G-CSF Mobilized CD34+ Selected Hematopoietic Precursor Cells Co-Infused With a Reduced Dose of Non-Mobilized Donor T-cells

Background: * Stem cell transplants from related donors (allogenic stem cell transplants) can be used to treat individuals with certain kinds of severe blood diseases or cancers, such as severe anemia. Allogenic stem cell transplants encourage the growth of new bone marrow to replace that of the recipient. Because stem cell transplants can have serious complications, researchers are interested in developing new approaches to stem cell transplants that will reduce the likelihood of these complications. * By reducing the number of white blood cells included in the blood taken during the stem cell collection process, and replacing them with a smaller amount of white blood cells collected prior to stem cell donation, the stem cell transplant may be less likely to cause severe complications for the recipient. Researchers are investigating whether altering the stem cell transplant donation procedure in this manner will improve the likelihood of a successful stem cell transplant with fewer complications. Objectives: \- To evaluate a new method of stem cell transplantation that may reduce the possibly of severe side effects or transplant rejection in the recipient. Eligibility: * Recipient: Individuals between 4 and 80 years of age who have been diagnosed with a blood disease that can be treated with allogenic stem cell transplants. * Donor: Individuals between 4 and 80 years of age who are related to the recipient and are eligible to donate blood. OR unrelated donors found through the National Marrow Donor Program. Design: * All participants will be screened with a physical examination and medical history. * DONORS: * Donors will undergo an initial apheresis procedure to donate white blood cells. * After the initial donation, donors will receive injections of filgrastim to release bone marrow cells into the blood. * After 5 days of filgrastim injections, donors will have apheresis again to donate stem cells that are present in the blood. * RECIPIENTS: * Recipients will provide an initial donation of white blood cells to be used for research purposes only. * From 7 days before the stem cell transplant, participants will be admitted to the inpatient unit of the National Institutes of Health Clinical Center and will receive regular doses of cyclophosphamide, fludarabine, and anti-thymocyte globulin to suppress their immune system and prepare for the transplant. * After the initial chemotherapy, participants will receive the donated white blood cells and stem cells as a single infusion. * After the stem cell and white blood cell transplant, participants will have regular doses of cyclosporine and methotrexate to prevent rejection of the donor cells. Participants will have three doses of methotrexate within the week after the transplant, but will continue to take cyclosporine for up to 4 months after the transplant. * Participants will remain in inpatient care for up to 1 month after the transplant, and will be followed with regular visits for up to 3 years with periodic visits thereafter to evaluate the success of the transplant and any side effects.