Inclusion Criteria:

- Understand and voluntarily sign an informed consent form

- Able to adhere to the study visit schedule and other protocol requirements

- Histologically confirmed AL or LCDD (from any time prior to screening)

- Up to one cycle of prior therapy is 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 weeks has to have had passed since last dose of melphalan, 2 weeks since
last velcade or glucocorticoid dose

- Measurable disease, as defined by:

- Serum monoclonal protein >= 0.5 g/dL by serum electrophoresis

- Urine monoclonal protein > 200 mg/tv in a 24 hour urine electrophoresis

- A difference between the involved immunoglobulin free light chain and uninvolved
light chain of >= 5 mg/dL AND abnormal serum immunoglobulin kappa lambda free
light chain ratio

- 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

- Absolute neutrophil count >= 1000/mm^3

- Platelet count >= 75,000/mm^3

- Serum creatinine =< 2.5 mg/dL

- Total bilirubin =< 1.5 mg/dL

- Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT]) and
alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 3 x
upper limit of normal (ULN)

- Disease free of prior malignancies for >= 5 years with exception of currently treated
basal cell, squamous cell carcinoma of the skin, or carcinoma "in situ" of the cervix
or breast

- Females of childbearing potential (FCBP) must have a negative serum or urine
pregnancy test with a sensitivity of at least 25 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

- Able to take aspirin (81 or 325 mg) daily as prophylactic anticoagulation (patients
intolerant to acetylsalicylic acid [ASA] may use warfarin or low molecular weight
heparin), unless baseline prothrombin time [PT] or partial thromboplastin time [PTT]
is >= 1.5 ULN, in which case thromboprophylaxis is not required

Exclusion Criteria:

- Any serious medical condition, laboratory abnormality, or psychiatric illness that
would prevent the subject from signing the informed consent form

- Pregnant or breast feeding females; (lactating females must agree not to breast feed
while taking pomalidomide)

- 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

- Use of any other experimental drug or therapy within 28 days of baseline

- Known hypersensitivity reaction or history of desquamating rash related to
thalidomide or lenalidomide

- Known hypersensitivity to bortezomib, boron, or any of the other agents utilized in
this protocol

- Patient has >= grade 3 peripheral sensory neuropathy or >= grade 2 painful sensory
neuropathy within 14 days before enrollment; (NOTE: patient with peripheral
neuropathy [PN] that was previously this severe but is currently improved due to
ongoing therapy [e.g., gabapentin or amitriptyline] may be eligible)

- Myocardial infarction within 6 months prior to enrollment or has New York Heart
Association (NYHA) class III or IV heart failure, uncontrolled angina, severe
uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute
ischemia or active conduction system abnormalities (not including 1st degree
atrioventricular (AV)-block, Wenckebach type 2nd degree heart block, or left bundle
branch block; prior to study entry, any electrocardiogram (ECG) abnormality at
screening has to be documented by the investigator as not medically relevant); note:
there is no lower limit of left ventricular ejection fraction below which patients
are excluded from participation

- Concurrent use of other anti-cancer agents or treatments

- Known positive for human immunodeficiency virus (HIV) or infectious hepatitis, type
A, B or C

- Meets criteria for symptomatic multiple myeloma, defined as:

- >= 30% monoclonal plasma cells in the marrow AND ANY OF THE FOLLOWING:

- Biopsy-confirmed plasmacytoma

- Lytic bone lesion(s)

- Hypercalcemia without other explanation

- NOTE: patients with >= 30% marrow plasma cells WITHOUT any other criteria to
suggest myeloma except for symptoms related to amyloidosis ARE potentially
eligible