Combination Chemotherapy With or Without Bortezomib in Treating Patients With Classical Hodgkin Lymphoma That Has Returned or Does Not Respond to Prior Treatment.

Inclusion Criteria: * Relapsed or refractory classical Hodgkin lymphoma. * Patients must have received a front-line standard anthracycline-containing regimen, such as adriamycin-bleomycin-vinblastine-dacarbazine (ABVD), Stanford V, or bleomycin-etoposide-adriamycin-cyclophosphamide-oncovin-procarbazine-prednisone (BEACOPP). * Bi-dimensionally measurable disease with at least 1 lesion \>= 2.0 cm in a single dimension. * Absolute neutrophil count (ANC) \>= 1,500/microL. * Platelet count \>= 100,000/ microL. * Hemoglobin \>= 8 g/dL. * Serum bilirubin \< 2.0 mg/dL. * Alkaline phosphatase \< 2 x upper limits of normal (ULN). * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \< 2 x ULN. * Serum creatinine =\< 1.5 mg/dL. * Eastern Cooperative Oncology Group (ECOG) performance status of less than or equal to 2. * Females of childbearing potential must have a negative serum beta-human chorionic gonadotropin (hCG) pregnancy test and must agree to use 2 highly effective contraceptive methods (hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) during the study and for 3 months after completion of protocol treatment. Females of non-childbearing potential are those who are postmenopausal for greater than 1 year or whom have had a bilateral tubal ligation or hysterectomy. * Males who have partners of childbearing potential must agree to use an effective contraceptive method during the study and for 3 months after completion of protocol treatment. * Voluntary written 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performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care. Exclusion Criteria: * Lymphocyte predominant Hodgkin lymphoma histology. * More than one prior chemotherapy regimen. * Prior autologous or allogeneic stem cell transplant. * Presence of central nervous system (CNS) involvement with Hodgkin lymphoma. * Known human immunodeficiency virus (HIV) infection or acquired immunodeficiency syndrome (AIDS). * Active hepatitis B or C infection or history of cirrhosis. * Grade 2 or greater peripheral neuropathy within 14 days of enrollment. * Hypersensitivity to boron or mannitol. * Prior bortezomib therapy. * Another primary malignancy (other than squamous cell and basal cell carcinoma of the skin, in situ carcinoma of the cervix, or squamous intraepithelial lesion on PAP smear, or treated prostate cancer with a stable prostate specific antigen \[PSA\]) for which the patient has not been disease-free for at least 3 years. * Patients with congestive heart failure, Class III or IV, by New York Heart Association (NYHA) criteria. * Patients with a myocardial infarction 6 months prior to enrollment, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiogram (ECG) evidence of acute ischemia or active conduction system abnormalities. * Patient with other medical or psychiatric illness that is likely to interfere with participation in this clinical study. * Female subject that is pregnant or breast-feeding. * Patient that has received other investigational drugs within 14 days of enrollment. * Patients using concurrent therapy with corticosteroids at greater than or equal to 20 mg/day of prednisone equivalent. * Patients with active systemic bacterial, viral, or fungal infections that have required IV antimicrobials within 4 weeks prior to protocol treatment.