Brief Summary: This phase I/II trial studies how well cytokine-treated veto cells work in treating patients with hematologic malignancies following stem cell transplant. Giving chemotherapy and total-body irradiation before a stem cell transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. When the healthy stem cells from a donor are infused into the patient, they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Cytokine-treated veto cells may help the transplanted donor cells to develop and grow in recipients without causing graft-versus-host-disease (GVHD - when transplanted donor tissue attacks the tissues of the recipient's body).

Detailed Description: PRIMARY OBJECTIVE: I. To determine the optimal dose of anti-viral veto cells, defined as the dose which achieves engraftment without severe graft-vs-host disease (GVHD) at 42 days after non-myeloablative megadose T cell depleted haploidentical hematopoietic cell transplantation (HCT). SECONDARY OBJECTIVES: I. Toxicity. II. Response rate. III. Time to progression. IV. Infections. V. Immune reconstitution. VI. Overall survival up to 1 year. OUTLINE: This is a dose-escalation study of cytokine-treated veto cells. CONDITIONING REGIMEN: Patients receive anti-thymocyte globulin (ATG) intravenously (IV) over 4 hours on days -9 to -7 and fludarabine IV over 1 hour on days -6 to -3, then undergo total body irradiation (TBI) on day -1. TRANSPLANT: Patients undergo peripheral blood stem cell transplantation (PBSCT) IV over 30-60 minutes on day 0. GVHD PROPHYLAXIS: Patients receive cyclophosphamide IV over 3 hours on days +3 and +4 and cytokine-treated veto cells IV over 30-60 minutes on day +7. After completion of stem cell transplant, patients are followed up once a week for 4 weeks, once a month for 3 months, and then periodically for one year.