This is a multi-institution, single arm, non-randomized pilot study coordinated by the Pediatric Blood and Marrow Transplant Consortium. Eligible patients will have severe combined immunodeficiency syndrome (SCID) or severe T-cell immunodeficiency disorder. Patients with these disorders do not have properly functioning immune systems. Without treatment, these disorders result in early childhood death. The standard treatment used for these diseases is to give the patient a stem cell transplant from a matched donor. The donor cells can be from a family member, an unrelated marrow donor or umbilical cord blood. The donor source will impact on transplant risks and approaches to the preparative regimen.