Venetoclax and Tocilizumab for the Treatment of Patients With Relapsed or Refractory t(11;14) Multiple Myeloma

Subject must be \>= 18 years of age Subject has an Eastern Cooperative Oncology Group (ECOG) performance score of =\< 2 Diagnosis of multiple myeloma that requires treatment and has been previously treated with: * Has received at least 2 prior lines of therapy * Has had documented disease progression on or within 60 days after completion of the last therapy. * Has received at least 2 consecutive cycles of lenalidomide and be relapsed/refractory to lenalidomide, as defined per protocol. * Has received at least 2 consecutive cycles of a proteasome inhibitor (PI). Have MM positive for t(11;14) translocation as determined by an analytically validated fluorescence in-situ hybridization (FISH) assay per the central laboratory testing Subject must have had measurable disease at Screening, defined as any of the following: * Serum monoclonal protein \>= 1.0 g/dL (\>= 10 g/L) by protein electrophoresis, or * \>= 200 mg of monoclonal protein in the urine on 24-hour electrophoresis, or * Serum immunoglobulin free light chain (FLC) \>= 10 mg/dL provided serum FLC ratio is abnormal Subjects with a history of autologous transplantation must have adequate peripheral blood counts as defined below, have recovered from any transplant related toxicity(s) and be \> 100 days post-autologous transplant (prior to first dose of study drug) Subjects must meet the following laboratory parameters, per laboratory reference range, at least once during the screening period: * Absolute neutrophil count (ANC) \>= 1000/uL (Subject may use granulocyte colony-stimulating factor \[G-CSF\] to achieve ANC eligibility criteria) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 1.5 x upper limit of normal range (ULN) * Calculated creatinine clearance \>= 30 mL/min using a modified Cockcroft- Gault calculation or a 24-hour urine collection for creatinine clearance * Platelet count \>= 75,000 cells/mm3 and independent of transfusion for 2 weeks * Hemoglobin \>= 8.0 g/dL, subjects may not receive blood transfusion within 1 week to achieve hemoglobin eligibility criteria per investigator discretion * Total bilirubin =\< 1.5 x ULN; subjects with Gilbert's syndrome may have bilirubin \> 1.5 x ULN If female, subject must be: * Postmenopausal defined as: * Age \> 55 years with no menses for 24 or more months without an alternative medical cause * Age =\< 55 years with no menses for 24 or more months without an alternative medical cause * AND an follicle stimulating hormone (FSH) level \> 40 IU/L. OR * Permanently surgical sterile (bilateral oophorectomy, bilateral salpingectomy, or hysterectomy) OR * A woman of childbearing potential (WOCP) practicing at least one protocol specified method of birth control starting at cycle 1 day 1 (or earlier) through at least 30 days after last dose of study drug * Females of childbearing potential (must have negative results for pregnancy test performed: * At screening, on a serum or urine sample obtained within 28 days prior to the first study drug administration, * Prior to dosing, on a urine sample obtained on the first day of study drug dosing, if it has been \> 7 days since obtaining the serum pregnancy test results * Females of non-childbearing potential (either postmenopausal or permanently surgically sterile as defined above) at screening do not require pregnancy testing Must voluntarily sign and date an informed consent, approved by an Independent Ethics Committee (IEC)/Institutional Review Board (IRB), prior to the initiation of any screening or study-specific procedures Exclusion Criteria: Subject exhibits evidence of other clinically significant uncontrolled condition(s), including, but not limited to: * Acute infection within 14 days prior to first dose of study drug requiring antibiotic, antifungal, or antiviral therapy * Diagnosis of fever and neutropenia within 1 week prior to first dose of study drug Subject has a cardiovascular disability status of New York Heart Association class \>= 3 Subject has a significant history of renal, neurologic, psychiatric, endocrinologic, metabolic, immunologic, cardiovascular or hepatic disease within the last 6 months that, in the opinion of the investigator, would adversely affect his/her participation in the study Subject has a history of other active malignancies other than multiple myeloma within the past 3 years prior to study entry, with the following exceptions: * Adequately treated in situ carcinoma of the cervix uteri, * Basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin, * Localized prostate cancer Gleason grade 6 or lower AND with stable prostate specific antigen (PSA) levels off treatment * Previous malignancy confined and surgically resected (or treated with other modalities) with curative intent Known human immunodeficiency viral (HIV) infection Active hepatitis B or C infection based on screening blood testing Subject is receiving other ongoing anti-myeloma therapy Subject has received any of the following within 7 days prior to the first dose of study drug: * Strong or moderate CYP3A inhibitors, or * Strong or moderate CYP3A inducers Subject has received any of the following within 14 days prior to the first dose of study drug or has not recovered to less than a grade 2 clinically significant adverse effect(s)/toxicity(s) of the previous therapy: any anti-myeloma therapy including chemotherapy, radiotherapy, or investigational therapy, including targeted small molecule agents Subject has received prior treatment with a BCL-2 family inhibitor Subject is pregnant, parturient, or breastfeeding; deprived of freedom by judicial or administrative decision; hospitalized and unable to provide consent, or otherwise unable to provide consent Subject has consumed grapefruit, grapefruit products, Seville oranges (including marmalade containing Seville oranges) or star fruit within 3 days prior to the first dose of study drug Subject has received immunization with live vaccine within 60 days of dosing Recent corticosteroid therapy at a cumulative dose equivalent to \> 140 mg of prednisone or a single dose equivalent to \>= 40 mg of dexamethasone within 2 weeks prior to the first dose of study drug Subject's decision to not divulge the race