Bone Marrow Transplant From Partially Matched Donors and Nonmyeloablative Conditioning for Blood Cancers (BMT CTN 0603)

Inclusion Criteria:

* Participants must be 21 to 70 years old; participants 1 to 21 years old are also eligible if they are ineligible for BMT CTN #0501 (NCT00412360)
* Donor must be at least 18 years of age
* Human leucocyte antigen (HLA) typing will be performed at high resolution (allele level) for the HLA-A, -B, Cw, DRB1, and -DQB1 loci. A minimum match of 5/10 is required. An unrelated donor search is not required for a person to be eligible for this study if the clinical situation dictates an urgent transplant. Clinical urgency is defined as 6 to 8 weeks from referral to transplant center or low likelihood of finding a matched, unrelated donor. The donor and recipient must be identical, as determined by high resolution typing, on at least one allele of each of the following genetic loci: HLA-A, HLA-B, HLA-Cw, HLA-DRB1, and HLA-DQB1. Fulfillment of this criterion shall be considered sufficient evidence that the donor and recipient share one HLA haplotype, and typing of additional family members is not required.
* Must have received cytotoxic chemotherapy within 3 months of the consent date (measured from the start date of chemotherapy)
* Acute leukemias (includes T lymphoblastic lymphoma) in the second or subsequent complete remission (CR)
* Burkitt's lymphoma in the second or subsequent CR
* Lymphoma
* Patients with adequate physical function as measured by the following:

  1. Heart: left ventricular ejection fraction at rest must be greater than or equal to 35%, or shortening fraction greater than 25%
  2. Liver: bilirubin less than or equal to 2.5 mg/dL and alanine aminotransferase (ALT), aspartate aminotransferase (AST), and alkaline phosphatase less than five times the upper limit of normal
  3. Kidney: serum creatinine within normal range for age, or if serum creatinine is outside the normal range for age, then kidney function (creatinine clearance or glomerular filtration rate (GFR) is greater than 40 mL/min/1.73m\^2
  4. Pulmonary: forced expiratory volume in one second (FEV1), forced vital capacity (FVC), and carbon monoxide diffusing capacity (DLCO) greater than 50% predicted (corrected for hemoglobin). If unable to perform pulmonary function tests, then oxygen (O2) saturation must be greater than 92% on room air.
  5. Performance status: Karnofsky/Lansky score greater than or equal to 60%

Exclusion Criteria:

* Have an HLA-matched, related, or 7 or 8/8 allele matched (HLA-A, -B, -Cw, -DRB1) related donor able to donate
* Had an autologous hematopoietic stem cell transplant in the 3 months before study entry
* Pregnant or breastfeeding
* Evidence of HIV infection or known HIV positive serology
* Current uncontrolled bacterial, viral, or fungal infection (i.e., currently taking medication with evidence of progression of clinical symptoms or radiologic findings)
* Prior allogeneic hematopoietic stem cell transplant
* History of primary idiopathic myelofibrosis