First-line Pomalidomide, Bortezomib, and Dexamethasone For AL Amyloidosis or LCDD

Inclusion Criteria: * Understand and voluntarily sign an informed consent form * Able to adhere to protocol requirements * Histologically confirmed AL or LCDD (any time prior to screening) * Up to 1 cycle of prior therapy allowed (maximum of 120 mg total dexamethasone (or equivalent amount of prednisone), 4 days of melphalan, and/or 4 doses of velcade; at least 4 wks has to have had passed since last dose of melphalan, 2 wks since last velcade or glucocorticoid dose * Measurable light chain elevation, as defined by: * A difference between the involved immunoglobulin free light chain and uninvolved light chain and uninvolved light chain (dFLC) of \>= 5 mg/dL AND abnormal serum immunoglobulin kappa lambda free light chain ratio * EXCEPTION: during the DOSE ESCALATION PORTION of the study only, a measurable M-protein (\>= 0.5 g/dL) on serum protein electrophoresis (SPEP) or a measurable urinary light chain (\>= 200 mg/24 hrs) by urine protein electrophoresis (UPEP) without a dFLC meeting the above criteria is acceptable; subjects without a dFLC \>= 5 mg/dL treated at the MTD will not count towards the expansion cohort * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 at study entry * Demonstrated clonal population of plasma cells in the bone marrow or positive immunohistochemical stain with anti-light chain anti-sera of amyloid fibrils * NTproBNP \< 8500 pg/mL * Absolute neutrophil count \>= 1000/mm\^3 * Platelet count \>= 75,000/mm\^3 * Serum creatin…