This phase II trial tests the safety and effectiveness of giving ipilimumab and nivolumab in the morning compared to other times of day in treating patients with melanoma that is stage IV or that cannot be removed by surgery (unresectable). Immunotherapy with monoclonal antibodies, such as ipilimumab and nivolumab, may help the body's immune system attack the tumor and may interfere with the ability of tumor cells to grow and spread. While some patients have impressive outcomes with both of these drugs, over 40% of patients do not experience any clinical benefit. Studies have shown that the time of day that vaccines and other therapies are given have had an impact on response and survival. It is not known, however, whether time of day has an impact on response to immune checkpoint inhibitors, such as ipilimumab and nivolumab. Giving ipilimumab and nivolumab earlier in the day compared to later in the day may improve response to treatment and survival in patients with stage IV or unresectable melanoma.