A Study of Whether Ide-cel (bb2121) Can Be Made From People With Multiple Myeloma Who Have Had a Hematopoietic Cell Transplant

Inclusion Criteria: Cohort 1: * Patient with myeloma who has received at least four prior lines of treatment having been exposed to an IMID, PI, and a CD38 monoclonal antibody and had measurable disease prior to salvage high dose melphalan autoHCT done within the prior 2 - 6 months. (Salvage melphalan/AutoHCT can count as the 4th line of treatment). * Measurable disease is defined by any of the following: * M-spike ≥ 0.5mg/dL * Urine m-spike ≥ 200mg/dL/24 hours * Involved Serum Free light chain ≥ 10mg/dL * Measurable plasmacytoma on imaging (≥ 1 lesion that has a single diameter ≥ 2 cm). * Bone marrow plasma cells ≥ 30% as determined by CD138 immunohistochemistry staining Cohort 2: * Patients with pathologically confirmed MM who have received at least 4 prior lines of treatment having been exposed to an IMID, PI, and a CD38 monoclonal antibody and have undergone an allo HCT for RRMM at any time in their history and have at least minimal residual disease by flow or NGS in the bone marrow at least 3 months after allo HCT. * Prior to Leukapheresis: * Greater than age 18. * Karnofsky performance ≥ 70. * Recovered to Grade 1 or baseline of any non-hematologic toxicities due to prior treatments, excluding Grade 2 neuropathy * Not receive any systemic anti-myeloma therapy for 14 days prior to leukapheresis. Therapeutic doses of corticosteroids (defined as greater than 10 mg/day prednisone or equivalent) are permitted until within 72 hours prior to Leukapheresis. * Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 without filgrastim use in the prior 14 days. * Hemoglobin ≥ 8 g/dL (without red blood cell transfusion in the previous 7 days) * Creatinine Clearance (CrCl) ≥ 45 mL/min, measured or estimated by Cockcroft-Gault equation. * Corrected serum calcium ≤ 13.5 mg/dL * Oxygen saturation ≥ 92% on room air * Hepatic Function: * Serum aspartate aminotransferase (AST) or alanine aminotransferase (ALT) ≤ 2.5 x upper limit of normal (ULN * Serum total bilirubin Serum total bilirubin ≤ 2 x ULN. Patients who have been diagnosed with Gilbert's disease are permitted to exceed the defined bilirubin value of 2 x ULN 2 x ULN. Patients who have been diagnosed with Gilbert's disease are permitted to exceed the defined bilirubin value of 2 x ULN * International ratio (INR) or partial thromboplastin time (PTT) ≤ 1.5 x ULN * Cardiac Function: left ventricular ejection fraction ≥ 45% by echocardiogram (ECHO) or multigated acquisition scan (MUGA). * Willing and able to adhere to the study visit schedule and other protocol requirements including regulatory requirement of a 15 year follow up using the CIBMTR long term follow up mechanism. * Female patients of childbearing potential (FCBP) must: * Have a negative serum pregnancy test with a sensitivity of at least 50 mIU/mL prior to enrollment * Agree to use, and be able to comply with, TWO acceptable methods of birth control, one highly effective method and one additional effective (barrier) method AT THE SAME TIME, from screening through at least 1 year following Ide-Cel infusion * Agree to abstain from breastfeeding from screening through at least 1 year following Ide-Cel infusion * Male patients must: * Agree to use a condom during sexual contact with a pregnant female or a FCBP, even if he has undergone a successful vasectomy, from screening through at least 1 year following Ide-Cel infusion * Must not donate sperm from screening through at least 1 year following Ide-Cel infusion * Prior to LD Chemotherapy * Females of childbearing potential must have a negative serum pregnancy test ≤ 7 days prior to LD chemotherapy * Platelet count ≥ 50,000/mm\^3 (transfusion allowed) * ANC ≥ 1,000/mm\^3 (without filgrastim within 72 hours) * Hepatic Function: * Serum AST and ALT ≤ 2.5 × ULN * Serum total bilirubin ≤ 2 × ULN. Patients who have been diagnosed with Gilbert's disease are permitted to exceed the defined bilirubin value of 2 x ULN * CrCl ≥ 45 mL/min, measured or estimated by Cockcroft-Gault equation * INR or PTT ≤ 1.5 x ULN * No history of ≥ Grade 2 hemorrhage within 30 days * No presence of active/uncontrolled infection requiring systemic therapy. Prophylactic antimicrobials are allowed. * No intercurrent illness or toxicity that would place the subject at undue risk of proceeding to LD chemotherapy * Must not be taking therapeutic doses of corticosteroids (defined as greater than 10 mg/day prednisone or equivalent) within 72 hours prior to LD chemotherapy. Physiologic replacement, topical, intranasal and inhaled steroids are permitted. Patients on calcineurin inhibitors (cyclosporine or tacrolimus) should have levels considered undetectable per institutional criteria * No active urinary outflow obstruction * Availability of manufactured cells * Patients not meeting these criteria may still be eligible to initiate LD chemotherapy with the approval of the Protocol PI (Principal Investigator). * Prior to Ide-Cel or Cilta-Cel infusion * Subjects who meet at least one of the following criteria on the day of scheduled CAR T cell infusion should have its administration delayed: * Suspected or active systemic infection * Onset of fever ≥ 38°C * Requirement for supplemental oxygen to keep saturation greater than 91% * Cardiac arrhythmia not controlled with medical management * Hypotension requiring vasopressor support * New onset or worsening of other non-hematologic organ dysfunction ≥ Grade 3 * Taking any of the prohibited medications as described in Section 15 * Significant worsening in clinical status compared to initial eligibility criteria that would, in the opinion of the treating physician, increase the risk of adverse events associated with Ide-Cel or Cilta-Cel infusion. Exclusion Criteria: * Receiving any of the following less than 14 days prior to enrollment: * Plasmapheresis * Major surgery (as defined by the investigator) * Radiation therapy other than local therapy for MM-associated bone lesions * Prior organ transplant requiring systemic immunosuppressive therapy * History of ≥ Grade 2 hemorrhage within 30 days of enrollment * Patient requiring ongoing treatment with chronic, therapeutic dosing of anticoagulants (e.g., Warfarin, low molecular weight heparin, Factor Xa inhibitors) can be enrolled with approval of the PI. * History or presence of clinically relevant CNS pathology such as epilepsy, seizure, paresis, aphasia, stroke, subarachnoid hemorrhage or other CNS bleed, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or psychosis * Having concurrent Waldenstrom's macroglobulinemia, POEMS (polyneuropathy, organomegaly, endocrinopathy, monoclonal protein, and skin changes), or clinically significant amyloidosis * History of Class III or IV congestive heart failure (CHF) or severe nonischemic cardiomyopathy, unstable or poorly controlled angina, myocardial infarction, or hemodynamically significant ventricular arrhythmia within the previous 6 months prior to enrollment * Active clinically significant autoimmune disease, defined as a history of requiring systemic immunosuppressive therapy and at ongoing risk for potential disease exacerbation. Patients with a history of autoimmune thyroid disease, asthma, or limited skin manifestations are potentially eligible. Patients with a history of acute or chronic GVHD are potentially eligible if on minimal immunosuppressants as defined previously. * Seropositive for human immunodeficiency virus (HIV-1), chronic or active hepatitis B or C, or acute hepatitis A. If any history of exposure to hepatitis B or C, then DNA PCR should be negative. If hepatitis B core Ab positive with negative DNA PCR, patients should be on prophylaxis while on study. * Prior malignancies except resected basal cell carcinoma or treated carcinoma in situ. Cancer treated with curative intent less than 5 years prior to enrollment will not be allowed unless approved by the PI. Cancer treated with curative intent greater than 5 years prior to enrollment is allowed. * Female patients who are breastfeeding or who intend to become pregnant during participation in the study. * Known allergy or hypersensitivity to any of the study medications, their analogues, or excipients in the various formulations of any agent. * Serious medical of psychiatric illness likely to interfere with participation on this clinical study * Uncontrolled bacterial, viral or fungal infections (currently taking medication and with progression or no clinical improvement) at time of enrollment. * Unwilling or unable to provide informed consent * Unable or unwilling to return to the center for treatment and follow up * No systemic anti-myeloma therapy is allowed within 7 days prior to leukapheresis. Steroids are allowed, but should be tapered off by 72 hours prior to leukapheresis. * Steroids are allowed between leukapheresis and LD chemotherapy, but should be tapered off by 72 hours prior to lymphodepletion.