Nivolumab Plus Ipilimumab in Thyroid Cancer

Inclusion * Metastatic, RAI refractory, differentiated thyroid cancer (including papillary and follicular thyroid cancer and their sub-types such as Hurthle cell thyroid cancer as well as poorly differentiated thyroid cancer), with progression within 13 months prior to study registration. RAI refractoriness is defined as absence of uptake of RAI on either a low-dose diagnostic test or a post-treatment RAI scan in measurable lesions or radiographic progression of disease within 12 months of the last course of RAI treatment despite the recorded uptake of RAI with that previous therapy or having a cumulative lifetime administered dose of greater than 600mCi. * Exploratory cohort: incurable medullary thyroid cancer with prior tyrosine kinase inhibitor (TKI) failure and progression within 13 months prior to enrollment (10 patients) and anaplastic thyroid cancer (7 patients) * Any number of lines of prior treatment are allowed * Any line of prior treatment for patients under 65y, over 65y must have at least one prior line of TKI treatment * Age 18 years or older * ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A) * Participants must have normal organ and marrow function as defined below: * Screening laboratory values must meet the following criteria and should be obtained within 21 days prior to randomization/registration * WBC ≥ 2000/μL * Neutrophils ≥ 1500/μL * Platelets ≥ 100 x103/μL * Hemoglobin \> 9.0 g/dL * Serum creatinine ≤ 1.5 x ULN or creatinine clearance (CrCl) ≥ 40 mL/min (if using the Cockcroft-Gault formula below): * Female CrCl = (140 - age in years) x weight in kg x 0.85 72 x serum creatinine in mg/dL * Male CrCl = (140 - age in years) x weight in kg x 1.00 72 x serum creatinine in mg/dL * AST/ALT ≤ 3 x ULN * Total Bilirubin ≤ 1.5 x ULN (except subjects with Gilbert Syndrome, who can have total bilirubin \< 3.0 mg/dL) * Ability to understand and the willingness to sign a written informed consent document. * Women of childbearing potential (WOCBP) must use appropriate method(s) of contraception. WOCBP should use an adequate method to avoid pregnancy for 23 weeks (30 days plus the time required for nivolumab to undergo five half-lives) after the last dose of investigational drug * Women of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 iu/l or equivalent units of hcg) within 24 hours of the first dose of the study drug * Women of childbearing potential" is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. In addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL. * Men who are sexually active with WOCBP must use any contraceptive method with a failure rate of less than 1% per year Men receiving nivolumab and who are sexually active with WOCBP will be instructed to adhere to contraception for a period of 31 weeks after the last dose of investigational product Women who are not of childbearing potential (ie, who are postmenopausal or surgically sterile as well as azoospermic men do not require contraception Exclusion * Patients should be excluded if they have an active, known or suspected autoimmune disease. Subjects are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger * Patients should be excluded if they have a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \> 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. * As there is potential for hepatic toxicity with nivolumab or nivolumab/ipilimumab combinations, drugs with a predisposition to hepatoxicity should be used with caution in patients treated with nivolumab-containing regimen. * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier. * Patients who are receiving any other investigational agents. * Patients with activebrain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * 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.