Stem Cell Transplant Using Peripheral and Cord Blood Stem Cells to Treat Severe Aplastic Anemia and Myelodysplastic Syndrome

* INCLUSION CRITERIA - RECIPIENT: * Diagnosed with severe aplastic anemia characterized by all of the following: 1. Bone marrow cellularity less than 30 percent (excluding lymphocytes) 2. Transfusion dependence for platelets and/or red blood cells (RBCs) 3. Neutropenia (absolute neutrophil count less than 500 cells/microL) OR for patients receiving granulocyte transfusions, absolute neutrophil count \< 500 cells/microL before beginning granulocyte transfusions\].) OR * Diagnosed with myelodysplastic syndrome characterized by refractory anemia OR refractory anemia with ringed sideroblasts (RARS and at least one of the following: 1. Neutropenia \[(absolute neutrophil count \< 500 cells/microL) OR for patients receiving granulocyte transfusions, absolute neutrophil count \< 500 cells/microL before beginning granulocyte transfusions\]) and history of 1 or more opportunistic infections related to neutropenia. OR 2. History of severe aplastic anemia transformed to MDS * Intolerance of or failure to respond standard immunosuppressive therapy. * Availability of at least one HLA-haploidentical (i.e. greater than or equal to 5/10 and less than or equal to 8/10 HLA match) related donor (HLA-A, B, C, DR, and DO loci) who is available to donate CD34+ cells (4-75 years old). * Availability of at least one 4/6 HLA-matched (HLA-A, B, and DR loci) cord blood unit from the National Marrow Donor Program (NMDP). The cord blood unit must contain a minimum total nucleated cells (TNC) (prior to thawing) of at least 1.5 x 10(7) cells per kilogram of recipient body weight with the following exception: if the minimum criterion of TNC is not met the cord unit must contain at least 1.7 x 10(5) CD34 plus cells/kg (prior to thawing). * Ages 4-55 years inclusive. * Ability to comprehend the investigational nature of the study and provide informed consent. The procedure will be explained to subjects aged 4-17 years with formal consent being obtained from parents or legal guardian. * Telomere Length Testing * In patients where a suspicion for a familial bone marrow failure syndromes (BMFS) exists, TERC and TERT mutation testing will be performed on protocol 04-H-0012 or performed elsewhere prior to enrolling on 04-H-0012. EXCLUSION CRITERIA - RECIPIENT: * Availability of an HLA identical or 9/10 HLA matched(HLA A, B, C, DR, and DO loci-relative to serve as a stem cell donor. * The patient is deemed to be a candidate for a 6/6 HLA matched unrelated stem cell transplant (availability of a donor and resources required for such a transplant). * Eastern Cooperative Oncology Group (ECOG) performance status of 2 or more. * Major anticipated illness or organ failure incompatible with survival from transplant * Severe psychiatric illness. Mental deficiency sufficiently severe as to make compliance with the transplant treatment unlikely and making informed consent impossible. * Positive pregnancy test for women of childbearing age. * HIV positive * Diagnosis of Fanconi anemia (by chromosome breakage study). * Diffusion capacity of carbon monoxide (DLCO) less than 40 percent using DLCO corrected for Hgb or lung volumes (patients under the age of 10 may be excluded from this criterion if they have difficulty performing the test correctly and thus are unable to have their DLCO assessed) using DL Adj and DL/VA/Adj. * Left ventricular ejection fraction less than 40 percent (evaluated by ECHO) * Transaminases greater than 5x upper limit of normal (when transaminases are elevated, the subject may be excluded at the discretion of the PI). * Serum bilirubin greater than 4 mg/dl * Creatinine clearance less than 50 cc/min by 24 hr urine collection (adjusted for body surface area, i.e.50 ml/min/1.73m(2)) * Serum creatinine \> 2.5 mg/dl * Failure to collect an adequate number of CD34+ cells (i.e. greater than or equal 2 x 10(6) CD34+ cells/kg) for transplantation from the subject s haploidentical relative. * Presence of an active infection not adequately responding to appropriate therapy * History of a malignant disease liable to relapse or progress within 5 years INCLUSION CRITERIA - RELATED HAPLOIDENTICAL DONOR DONATING PURIFIED CD34 PLUS CELLS: * HLA mismatched family donor (greater than or equal to 5/10 and less than or equal to 8/10 HLA matched (HLA-A, B, C, DR and DO loci) who is available to donate CD34+ cells. * Ages 4-75 inclusive * Weight greater than or equal to 15 kg. * For adults: Ability to comprehend the investigational nature of the study and provide informed consent. For minors: Written informed consent from one parent or guardian who is not the recipient of the transplant and informed assent: The process will be explained to the minor on a level of complexity appropriate for their age and ability to comprehend. * If there is a suspicion of familial BMFS in the recipient, then the donor must have undergone genetic testing for genes associated with BMFS - performed at a CLIA-certified laboratory, prior to enrolling in this protocol. EXCLUSION CRITERIA RELATED DONOR (ANY OF THE FOLLOWING): * Pregnant or breastfeeding. * A suitable adult haplo identical donor is available. * Unfit to receive filgrastim (G-CSF) and undergo apheresis (history of stroke, MI, unstable angina, uncontrolled hypertension, severe heart disease or palpable spleen). * HIV positive (Donors who are positive for HBV, HCV or HTLV I/II, T. cruzi \[Chagas\] may be used at the discretion of the investigator following counseling and approval from the recipient). * Sickling hemoglobinopathies including HbSS, HbAS, HbSC. Donors with HbAS are acceptable. * Severe psychiatric illness. Mental deficiency sufficiently severe as to make compliance with the BMT treatment unlikely and making informed consent impossible. * Screening test positive for Chagas disease (Trypanosoma cruzi /T. cruzi/trypanosomiasis) confirmed by the Center for Disease Control (CDC).