Bortezomib and Chemotherapy in Treating Participants With Lymphoid Malignancies Undergoing Stem Cell Transplant

Inclusion Criteria:

* Any histological subtype of CD20+ lymphoid malignancies or T-cell lymphoid malignancies.
* Patients with CD20+ lymphoid malignancies in relapse after failing \>= 1 prior regimen of conventional treatment and not eligible for non-myeloablative transplant. Patients with T-cell lymphoid malignancies can either be in relapse or newly diagnosed with high risk features (such as high International Prognostic Index \[IPI\] of \>= 2).
* Patients with prior non-myeloablative transplant are eligible if not from the same donor.
* A fully-matched or one-antigen mismatched sibling or unrelated donor.
* Left ventricular ejection fraction (EF) \>= 40% with no uncontrolled arrhythmias or symptomatic heart disease.
* Forced expiratory volume in one second (FEV1), forced vital capacity (FVC) and diffusion capacity of the lung for carbon monoxide (DLCO) \>= 40%.
* Serum creatinine \< 1.8 mg/dL.
* Serum bilirubin \< 3 X upper limit of normal.
* Serum glutamate pyruvate transaminase (SGPT) \< 3 X upper limit of normal.
* Voluntary signed, written Institutional Review Board (IRB)-approved informed consent before performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care.
* Men and women of reproductive potential must agree to follow accepted birth control methods for the duration of the study. Female subject is either post-menopausal or surgically sterilized or willing to use an acceptable method of birth control (i.e., a hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study. Male subject agrees to use an acceptable method for contraception for the duration of the study.

Exclusion Criteria:

* Past history of anaphylaxis following exposure to rituximab or Velcade, boron or mannitol.
* History of grade 3 or 4 National Cancer Institute (NCI) toxicity with prior Velcade therapy.
* Patient with active central nervous system (CNS) disease.
* Pregnant (positive beta human chorionic gonadotropin \[HCG\] test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization) or currently breast feeding. Pregnancy testing is not required for post-menopausal or surgically sterilized women.
* Known infection with human immunodeficiency virus (HIV), human T-lymphotropic virus (HTLV-I), hepatitis B, or hepatitis C.
* Patients with other malignancies diagnosed within 2 years prior to study day -13 (except skin squamous or basal cell carcinoma).
* Active uncontrolled bacterial, viral or fungal infections.
* Major surgical procedure or significant traumatic injury within 4 weeks prior to day -13.
* Serious, non-healing wound, ulcer, or bone fracture.
* History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 3 months prior to day -13.
* History of stroke within 6 months.
* Myocardial infarction within the past 6 months prior to study day 1, or has New York Heart Association (NYHA) class III or IV heart failure or arrhythmia, unstable angina, uncontrolled congestive heart failure or arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities. Prior to study entry, any electrocardiography (ECG) abnormality at screening must be documented by investigator as not medically relevant.
* Uncontrolled hypertension (\>= 140/90).
* Uncontrolled chronic diarrhea.
* A prior allogeneic transplant from the same donor.
* Serious medical or psychiatric illness likely to interfere with participation in this clinical study.
* Patient has received other investigational drugs within 3 weeks before enrollment.
* Active peripheral neuropathy greater or equal to grade 2.