This phase II trial studies the efficacy (the effect on the tumor) and the safety (the effect on the body) of the study drugs when given as a combination in participants with this type of cancer. Another purpose of the study is to see which tumor markers (proteins in the blood that the body produces in response to the cancer) lead to better results in participants treated with the study drugs. Nivolumab and ipilimumab are antibodies, which are human proteins that recognize and attach to a part of the tumor and/or body's immune cells. They work in slightly different ways to activate the immune system and help the body's immune system to work against tumor cells. Nivolumab and ipilimumab are investigational because they are not approved by the FDA to be used for the type of cancer being studied.