Abatacept, Ixazomib Citrate, and Dexamethasone in Treating Patients With Multiple Myeloma Resistant to Chemotherapy

Inclusion Criteria: * Patients with multiple myeloma who have relapsed (or who are primary refractory) following treatment with a proteasome inhibitor-containing regimen (excluding ixazomib) and who have not been treated with a second proteasome inhibitor (ixazomib, bortezomib, carfilzomib or other proteasome inhibitor). * Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 at study entry * Must be free of systemic infection: * Subjects with active infections (whether or not they require antibiotic therapy) may be eligible after complete resolution of the infection * Subjects on antibiotic therapy must be off antibiotics for at least 7 days before beginning treatment * Absolute neutrophil count \>= 750/mm\^3 * Platelet count \>= 25,000/mm\^3 * Creatinine clearance \>= 30 mL/min * Total bilirubin =\< 3 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x ULN * Patient's multiple myeloma cells are positive for CD28 or CD86 expression by flow cytometry or immunohistochemistry (in any proportion) CD28 or CD86 positivity can have been determined on previous bone marrow aspirates or biopsies * Disease free of prior malignancies for \> 2 years with exception of currently treated basal cell, squamous cell carcinoma of the skin, or carcinoma ?in situ? of the cervix or breast * Participants of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure Exclusion Criteria: * Prior treatment with ixazomib * Inability to take ixazomib or abatacept * Life expectancy less than 4 months * Patients with a known diagnosis of plasma cell leukemia * Known active tuberculosis or fungal infection * Known seropositive for or active viral infection with, human immunodeficiency virus (HIV), hepatitis B virus (HBV) or hepatitis C virus (HCV) * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * 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, which confounds the ability to interpret data from the study * Pregnant or nursing female participants * Unwilling or unable to follow protocol requirements * Any condition which in the investigator?s opinion deems the participant an unsuitable candidate to receive study drug * Received an investigational agent within 30 days prior to enrollment