Pomalidomide, Bortezomib, and Dexamethasone in Treating Patients With Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Serum Creatinine =\< 3 mg/dL * Absolute neutrophil count \>= 1000/uL * Platelet count \>= 75,000/uL * 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) * measurable plasmacytoma that has not been radiated * ECOG Performance status (PS) 0, 1, or 2 * Previously treated relapsed or refractory multiple myeloma * Patients must have received at least one prior therapy but no more than 4 therapies. * one or more of the prior regimens must have included lenalidomide and it has been determined the patient is refractory, resistant, or relapsed this therapy * prior bortezomib not required but if prior exposure, patients should not be refractory (Refractory means progression on therapy or within 60 days from the last dose of bortezomib.) * Provide informed written consent * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 50 mIU/mL within 10 - 14 days prior to and again within 24 hours of starting pomalidomide 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 pomalidomide; 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 vasectomy * All patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * Willing and able to take aspirin or alternate prophylactic anticoagulation * All previous cancer therapy, including chemotherapy, high dose corticosteroids, immune modulatory drugs or proteosome inhibitors must have been discontinued \>= 2 weeks prior to study registration * Any prior treatment with investigational agents must be discontinued \>= 28 days prior to study registration * Willing to abstain from donating blood during study participation and for 28 days after discontinuation of study drug * Men must agree to abstain from donating semen or sperm during study participation and for 28 days after discontinuation of study drug * Willingness to return to enrolling institution for follow-up Exclusion Criteria: * Concomitant high dose corticosteroids (concurrent use of corticosteroids); EXCEPTION: Patients may be on chronic steroids (maximum dose 20 mg/day prednisone equivalent) if they are being given for disorders other than myeloma, i.e., adrenal insufficiency, rheumatoid arthritis, etc * Another malignancy undergoing active treatment with the exception of non melanoma skin cancer or in situ cervical or breast cancer * Pregnant women or women of reproductive ability who are unwilling to use effective contraception * Nursing women * Men who are unwilling to use a condom (even if they have undergone a prior vasectomy) while having intercourse with any woman, while taking the drug and for 4 weeks after stopping treatment * 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 * Active DVT or PE that has not been therapeutically anticoagulated * Known positive for HIV or active hepatitis infection * Any condition, including the presence of laboratory abnormalities, which places the subject at unacceptable risk if he/she were to participate in the study or confounds the ability to interpret data from the study * Known hypersensitivity to thalidomide or lenalidomide including development of erythema nodosum if characterized by a desquamating rash * \> grade 2 peripheral neuropathy * Any prior use of pomalidomide