Bone Marrow Transplantation With Specially Treated Bone Marrow in Treating Patients With Hematologic Cancer That Have Not Responded to Previous Therapy

Inclusion Criteria: * Age ≤40 years. * Diagnoses-patients with the following hematologic malignancies and bone marrow failure syndromes: * Acute myelogenous leukemia-induction failure, relapse, second or greater complete remission (CR) * Acute lymphocytic leukemia-induction failure, relapse, second or greater CR, first CR with t(9;22), t(8;14), or t(4;11) * Non-Hodgkin's lymphoma (intermediate or high grade) which has failed to achieve CR with at least two induction regimens, relapse, second or greater CR * Multiple myeloma with poor prognostic features (elevated 0-2 microglobulin or high labeling index) * Hodgkin's disease in relapse or which fails to achieve CR after two chemotherapy regimens * Congenital or acquired bone marrow failure - poorly responsive to or intolerant of current therapy * Myelodysplastic syndrome of all subtypes except refractory anemia (RA) * Patient has a haploidentical family member that meets medical criteria for donation. * Eligibility for other transplant types: * Patient considered likely to have clinical deterioration and rapid disease progression during an unrelated donor search, or * Patient who has already had an unproductive donor search or * Patient ineligible for or has refused autologous transplant * Adequate renal and hepatic function for age: * Serum creatinine \<2 x ULN * Alanine aminotransferase (ALT, SGPT) x ULN * Aspartate aminotransferase (AST, SGOT) x ULN * Total bilirubin 5\_2 x ULN except if bilirubin is elevated due to Gilbert's syndrome or hemolytic anemia * Adequate cardiac and pulmonary function for age. * ECOG Performance Status 0, 1, or 2 or Lansky performance scale \>50% for patients \<16 years of age. * Voluntary witnessed written informed consent. Children will be asked for assent where appropriate. * The patient, if female, must be post-menopausal, premenarcheal, or sterile, or if the patient is of childbearing potential, she must be practicing a method of birth control considered effective and medically acceptable by the investigator for a minimum of 1 month prior to study entry and at least 2 months after the study end. * Patient must have undergone successful leukapheresis to obtain adequate antigen presenting cells. * Any patient who enters the study in a relapse state, with evidence of end organ (pulmonary, renal, or hepatic) toxicity, or with recent recovery from infection, who may potentially have little benefit from this protocol, must have his/her eligibility status discussed with the Principal Investigator. * Patient must have life expectancy of at least 12 weeks. Exclusion Criteria * Eligibility for other transplant types: * Patient has family donor who is matched or single antigen mismatched at HLA-A, HLA-B, HLA-DR, and HLA-DQ. Donorrecipient matching must be evaluated via both phenotype and genotype. * Patient has available unrelated donor who is matched at HLA-A, HLA-B, and HLA-DR. Donor-recipient matching must be evaluated via both phenotype and genotype. * Active uncontrolled infection (continued positive blood or soft tissue cultures despite appropriate antibiotic treatment) * Positive 13-HCG in a female of childbearing potential * Evidence of HIV infection or known HIV positive serology * Any prior bone marrow transplant * A peripheral blood differential count at the time of leukapheresis with greater than 25% blasts. This exclusion criterion is valid only for the first four patients enrolled. * Patients with Fanconi's anemia