Pomalidomide or Lenalidomide and Dexamethasone in Treating Patients With Relapsed or Refractory Multiple Myeloma Previously Treated With Lenalidomide

Inclusion Criteria: * Calculated creatinine clearance \>= 30 ml/min by Cockcroft-Gault formula * Absolute neutrophil count \>= 1000uL * (Untransfused) platelet count \>= 50000/uL * Hemoglobin \>= 8.0 g/dL * Relapsed myeloma that previously became refractory to lenalidomide, after initial response of partial response or better to the drug; refractory is defined as progression on treatment with a dose of at least 10 mg daily for lenalidomide; greater than or equal to 180 days must have elapsed since previous lenalidomide therapy was stopped * Measurable disease of multiple myeloma as defined by at least ONE of the following: * Serum monoclonal protein \>= 1.0 g/dL * \>= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis * Serum immunoglobulin free light chain \>= 10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio * Monoclonal bone marrow plasmacytosis \>= 30% (evaluable disease) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Previously treated; NOTE: no limit to prior therapy provided there is adequate residual organ function * Provide informed written consent * Females of childbearing potential (FCBP)\* must have a negative serum pregnancy test with a sensitivity of at least 50 mIU/mL within 10 - 14 days prior to and again within 24 hours prior to prescribing lenalidomide for cycle 1 (prescriptions must be filled within 7 days as required by Revlimid Risk Evaluation and Mitigation Strategy \[REMS\]), and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking lenalidomide; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a successful vasectomy; all study participants must be registered into the Revlimid REMS program, and be willing and able to comply with the requirements of Revlimid REMS program * A female of childbearing potential is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Willing to return to Mayo Clinic enrolling institution for follow-up Exclusion Criteria: * Residual toxicity of \> grade 1 from prior therapy * Other active malignancy \< 1 year prior to registration; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer * Any of the following: * Pregnant women * Nursing women (lactating females must agree not to breast feed while taking lenalidomide) * Men or women of childbearing potential who are unwilling to employ adequate contraception (condoms, diaphragm, birth control pills, injections, intrauterine device \[IUD\], or abstinence, etc.) * Other co-morbidity which would interfere with patient's ability to participate in trial, e.g. uncontrolled infection, uncompensated heart or lung disease * Other concurrent chemotherapy, radiotherapy, or any ancillary therapy considered investigational; NOTE: bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment * New York Heart Association classification III or IV * Diagnosed active deep vein thrombosis (DVT) that has not been therapeutically anticoagulated