Bortezomib in Treating Patients With Metastatic Thyroid Cancer That Did Not Respond to Radioactive Iodine Therapy

Inclusion Criteria: * The Eastern Cooperative Oncology Group (ECOG) 0-2 OR Karnofsky 60-100% * Platelet count \>= 100,000/mm\^3 * Absolute neutrophil count \>= 1,500/mm\^3 * White Blood Count (WBC) \>= 3,000/mm\^3 * Aspartate aminotransferase (AST) and Alanine aminotransferase (ALT) =\< 2.5 times upper limit of normal * Bilirubin normal * No symptomatic congestive heart failure * Creatinine normal OR creatinine clearance \>= 60 mL/min * No unstable angina pectoris * No cardiac arrhythmia * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No other malignancy within the past 5 years except basal cell skin cancer or carcinoma in situ of the cervix * No ongoing or active infection * No psychiatric illness or social situation that would preclude study compliance * No other uncontrolled illness * At least 4 weeks since prior chemotherapy * No more than 2 prior chemotherapy regimens * At least 6 months since prior external beam radiotherapy for locoregional disease in the thyroid bed or to the cervical or upper mediastinal lymph nodes (dose =\< 6,000 cGy) * At least 6 months since prior radioiodine therapy * No prior external radiotherapy to the measured tumor * Prior thyroidectomy allowed * No concurrent combination antiretroviral therapy for HIV-positive patients * No other concurrent investigational agents * No other concurrent anticancer therapy * Unresponsive to prior radioiodine therapy * Histologically confirmed differen…