Autologous Peripheral Blood Stem Cell Transplant Followed by Donor Bone Marrow Transplant in Treating Patients With High-Risk Hodgkin Lymphoma, Non-Hodgkin Lymphoma, Multiple Myeloma, or Chronic Lymphocytic Leukemia

Inclusion Criteria: * Must have the capacity to give informed consent * Detectable tumor prior to mobilization regimen * Patients with stored autologous stem cells will be allowed * Stem cells from an identical donor could be used for autologous hematopoietic cell transplant (HCT) * Marrow is the preferred source of stem cells from the HLA-haploidentical donor, however, peripheral blood mononuclear cells (PBMC) could be used as stem cell source, after clearance with the Fred Hutchinson Cancer Research Center (FHCRC) principal investigator, in the case of difficulties or contraindications to bone marrow harvest from the donor * Cross-over to other tandem autologous-allogeneic research protocol (#1409 or other appropriate protocol) will be allowed if a suitable HLA-matched related or unrelated donor is identified before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study * Cross-over from other tandem autologous-allogeneic research protocol (#1409 or other appropriate protocol) will be allowed if the patient loses the suitable HLA-matched related or unrelated donor but has an available HLA-haploidentical donor before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study * Lymphoma: patients with * Diagnosis of non-Hodgkin lymphoma (NHL) or Hodgkin's lymphoma (HL), of any histological grade * Refractory or relapsed disease after standard chemotherapy * High risk of early relapse following autograft alone * Waldenstrom's macroglobulinemia: must have failed 2 courses of therapy * CLL: * Patients with either a: * Diagnosis of T-cell CLL or T-cell prolymphocytic leukemia (PLL) who have failed initial chemotherapy, patients with T cell CLL or PLL or * Diagnosis of B-cell CLL, B-cell small lymphocytic lymphoma, or B-cell CLL that progressed to PLL who either: 1. Failed to meet National Cancer Institute (NCI) Working Group criteria for complete or partial response after therapy with a regimen containing fludarabine (or another nucleoside analog, e.g. 2-chlorodeoxyadenosine \[CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing fludarabine (or another nucleoside analog) 2. Failed any aggressive chemotherapy regimen, such as fludarabine, cyclophosphamide and rituximab (FCR), at any time point 3. Have "17p deletion" cytogenetic abnormality and relapsed at any time point after initial chemotherapy * Harvesting criteria for autologous HCT: * Previously collected PBMC may be used * Circulating CLL cells \< 5000 * Marrow involvement with CLL cells \< 50% * Multiple myeloma (MM): patients who * Have received induction therapy for a minimum of 4 cycles * In addition, patients must meet at least one of the following criteria I-IX (I-VII at time of diagnosis or pre-autograft): * Any abnormal karyotype by metaphase analysis except for isolated t(11,14), * Fluorescent in situ hybridization (FISH) translocation 4:14, * FISH translocation 14:16, * FISH deletion 17p, * Beta2 (B2)-microglobulin \> 5.5 mg/ml, * Cytogenetic hypodiploidy * Plasmablastic morphology (\>= 2%) * Recurrent or non-responsive (less than partial remission \[PR\]) MM after at least two different lines of conventional chemotherapy * Progressive MM after a previous autograft (provided stored autologous cluster of differentiation \[CD\]34 cells are available) * Plasma cell leukemia: after induction chemotherapy * DONOR: Related donors who are genotypically identical for one HLA haplotype and who may be mismatched at the HLA-A, -B, -C or DRB1 loci of the unshared haplotype with the exception of single HLA-A, -B or -C allele mismatches * DONOR: Marrow is the preferred source of stem cells from the HLA-haploidentical donor, however PBMC could be used as stem cell source, after clearance with the FHCRC principal investigator, in the case of difficulties or contraindications to bone marrow harvest from the donor * DONOR: In the case that PBMC will be used as stem cell source, ability of donors \< 18 years of age to undergo apheresis without use of a vascular access device; vein check must be performed and verified by an apheresis nurse prior to arrival at the Seattle Cancer Care Alliance (SCCA) * DONOR: Age \>= 12 years of age Exclusion Criteria: * Life expectancy severely limited by disease other than malignancy * Seropositive for the human immunodeficiency virus * Female patients who are pregnant or breastfeeding * Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Patients with active non-hematological malignancies (except non-melanoma skin cancers) or those with non-hematological malignancies (except non-melanoma skin cancers) who have been rendered with no evidence of disease, but have a greater than 20% chance of having disease recurrence within 5 years * This exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * Patients with fungal infection and radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Symptomatic coronary artery disease or ejection fraction \< 40% or other cardiac failure requiring therapy (or, if unable to obtain ejection fraction, shortening fraction of \< 26%); ejection fraction is required if the patient has a history of anthracyclines or history of cardiac disease; patients with a shortening fraction \< 26% may be enrolled if approved by a cardiologist * Corrected diffusion capacity of the lungs for carbon monoxide (DLCO) \< 50% of predicted, forced expiratory volume in one second (FEV1) \< 50% of predicted, and/or receiving supplementary continuous oxygen; the FHCRC principal investigator (PI) of the study must approve of enrollment of all patients with pulmonary nodules * Patient with clinical or laboratory evidence of liver disease will be evaluated for the cause of liver disease, its clinical severity in terms of liver function, bridging fibrosis, and the degree of portal hypertension; the patient will be excluded if he/she is found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL, and symptomatic biliary disease * Karnofsky score \< 50% for adult patients * Lansky play-performance score \< 40 for pediatric patients * Patient with poorly controlled hypertension despite multiple antihypertensives * DONOR: Donor-recipient pairs in which the HLA-mismatch is only in the host-versus-graft (HVG) direction * DONOR: Infection with human immunodeficiency virus (HIV) * DONOR: Weight \< 20 kg * DONOR: A positive anti-donor cytotoxic crossmatch