Donor Stem Cell Transplant in Treating Young Patients With Myelodysplastic Syndrome, Leukemia, Bone Marrow Failure Syndrome, or Severe Immunodeficiency Disease

OBJECTIVES: * Determine the efficacy and toxicity of stem cell-enriched, T-cell-depleted, haplocompatible allogeneic hematopoietic stem cell transplantation in children with high-risk myelodysplastic syndromes, high-risk leukemia, severe acquired or congenital cytopenias, or primary immunodeficiency diseases. * Determine the toxicity of a fludarabine and thiotepa-containing regimen in combination with lower doses of antithymocyte globulin in these patients. * Determine the engraftment rate in patients treated with this regimen. * Define T-cell reconstitution in these patients. * Determine the toxicity and effects of administering stem cell and T-cell boosts after transplantation on hematopoiesis and immune reconstitution in these patients. OUTLINE: This is a dose-escalation study of donor CD3+ cells and antithymocyte globulin (ATG). * Cytoreductive regimen: Patients undergo total body irradiation twice daily on days -9 to -7. Patients also receive fludarabine IV on days -6 to -2, thiotepa IV every 12 hours on day -6, and ATG IV on days -5 to -2. * Transplantation: Patients undergo CD34-enriched, T-cell-depleted, haplocompatible allogeneic peripheral blood stem cell or bone marrow transplantation on day 0. * Donor T-cell infusion:Patients with no active graft-vs-host disease and evidence of engraftment but low absolute CD34+ lymphocyte count at 12 weeks post transplant may receive donor CD3+ cells at 4-week intervals. * Donor stem cell boost: Patients with engraftment but either cytokine or transfusion dependent at 12 weeks post transplant may receive a boost of donor CD34+ cells. Cohorts of 3-6 patients receive escalating doses of donor CD3+ cells and ATG until the optimum is determined. The optimum dose is defined as the dose at which both engraftment and T-cell recovery occur, without dose-limiting toxicity, in ≥ 5 of 6 patients. After the completion of study treatment, patients are followed periodically for 5 years and then every 5 years thereafter. PROJECTED ACCRUAL: A total of 21 patients will be accrued for this study.