Donor Stem Cell Transplant in Treating Patients With Previously Treated Lymphoma, Multiple Myeloma, or Chronic Lymphocytic Leukemia

Inclusion Criteria: * Donors will be \<55 years of age and in good health as approved by the National Marrow Donor Program (NMDP) donor and collection centers. Related donors will be \< 70 years of age. * Recipients will be \<55 years, will have satisfactory organ function (excluding bone marrow) and will have a Karnofsky activity assessment \>90% and will have: * Creatinine \<2.0 mg/dl. * Bilirubin, Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<2 x normal. * Pulmonary function test and Carbon Monoxide Diffusing Capacity (DLCO) \> 50% of normal. * Multi Gated Acquisition Scan (MUGA) \>45% injection fraction. * Recipients with unrelated donor matched at the HLA A, B, DRBI loci, or if \< 35 years mismatched at a single HLA A or B, or DRBI locus. * Umbilical cord blood (5) used as an unrelated stem cell source will provide \> 2.0 x 10\^7 cells/kg and will be matched at 4 - 6 of 6 HLA A, B, and DRBI loci. Cord blood grafts may include a single or pair of cord units depending on the cell dose. * Partially matched related donors will be at least haploidentical (matched at \>3 of 6 HLA A, B, DRB1 loci). * Recipients will fall under one of the following disease categories * Chronic lymphocytic leukemia -- must have all three: * Rai Stage III/IV * Progression after previous Complete Response (CR) or Partial Response (PR) including purine antagonist (i.e. fludarabine). * Recent chemotherapy responsiveness * Advanced non-Hodgkin's lymp…