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 differentiated thyroid cancer-papillary or follicular type, including, but not limited to, any of the following variants: hurthle cell (oxyphilic), insular, columnar cell, tall cell * Metastatic disease * At least 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques OR \>= 10 mm by spiral CT scan * No prior radiotherapy to the only measurable lesion * No radioiodine uptake in the measured metastatic tumor by radioiodine scan (Note: Must have had \>= 1 radioiodine scan since the last radioiodine treatment) * No known brain metastases