Can a One-Two punch of immunotherapy and radiation tame sarcoma before surgery?

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: * Diagnosed with biopsy-proven soft-tissue sarcoma with confirmatory pathology review at University of Rochester Medical Center (URMC) * 18 years or older. * Able to provide informed consent. * Resectable disease per surgical evaluation. * Neoadjuvant/preoperative radiotherapy has been recommended * Intermediate to high grade sarcoma on biopsy, tumor \> 5 cm in size by imaging * Willing to have blood draws for flow cytometry and Serametrix analysis. * Willing to receive neoadjuvant radiation therapy and subsequent surgical resection. * Patients with known human immunodeficiency virus (HIV) are eligible if the lymphocytes \> 350 CD4+ cells and no detectable viral load. * Women of childbearing potential (defined as any woman, who 1) has not undergone tubal ligation, a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months or has had menses at any time in the preceding 24 consecutive months): * Must not be pregnant or nursing * Must have a negative pregnancy test done within 7 days prior to registration as well as within 72 hrs. prior to receiving first dose of study medication * Women of childbearing potential must use at least two other accepted and highly-effective methods of contraception and/or agree to abstain from sexual intercourse for at least 5 months after the last dose of nivolumab and/or ipilimumab. * Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. * Sexually active males must use at least two other accepted and highly-effective methods of contraception and/or agree to abstain from sexual intercourse for at least 7 months after the last dose of nivolumab and/or ipilimumab * Eastern Cooperative Oncology Group (ECOG) performance status 0 - 2 Exclusion Criteria: * Gastrointestinal stromal tumor, dermatofibrosarcoma protuberans, rhabdomyosarcoma, Ewing's sarcoma, low-grade sarcomas, osteosarcoma subtypes * History of radiation to the affected area * Evidence of metastatic disease prior to treatment * Any history of prior therapy with ipilimumab or nivolumab, or any agent targeting PD-1, PD-L1 or CTLA-4. * History of any the following: * Active known or suspected autoimmune disease * Active autoimmune colitis * Autoimmune pan hypopituitarism * Autoimmune adrenal insufficiency * Known active hepatitis B or C * Known active pulmonary disease with hypoxia defined as: * Oxygen saturation \< 85% on room air or * Oxygen saturation \< 88% despite supplemental oxygen * No systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration. * Second active malignancy, not including localized basal cell skin cancer, squamous cell skin cancer, in situ cervical cancer, ductal or lobular carcinoma in situ of the breast; patients with a history of other malignancies are eligible if they have been continuously disease-free for \> 3 years prior to the time of registration. * Absolute neutrophil count (ANC) \<= 1,500/mm\^3 * Platelet count \<= 100,000/mm\^3 * AST (Aspartate Aminotransferase) or ALT (Alanine Aminotransferase) =\> 2 x upper limit of normal (ULN) * Thyroid stimulating hormone (TSH) outside of normal limits; supplementation is acceptable to achieve a TSH within normal limits; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible. * Life expectancy under 5 years.