 This phase I trial studies the side effects and best dose of pomalidomide and bortezomib when given together with dexamethasone in treating patients with amyloid light-chain amyloidosis or light chain deposition disease. Biological therapies, such as pomalidomide, may stimulate the immune system in different ways and stop abnormal cells from growing. Bortezomib may stop the growth of abnormal cells by blocking some of the enzymes needed for cell growth. Giving pomalidomide and bortezomib together with dexamethasone may be an effective treatment for amyloid light-chain amyloidosis or light chain deposition disease 