Chordoma

This phase II trial studies nivolumab and ipilimumab in treating patients with rare tumors. Immunotherapy with monoclonal antibodies, such as nivolumab and ipilimumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. This trial enrolls participants for the following cohorts based on condition: 1. Epithelial tumors of nasal cavity, sinuses, nasopharynx: A) Squamous cell carcinoma with variants of nasal cavity, sinuses, and nasopharynx and trachea (excluding laryngeal, nasopharyngeal cancer \[NPC\], and squamous cell carcinoma of the head and neck \[SCCHN\]) B) Adenocarcinoma and variants of nasal cavity, sinuses, and nasopharynx (closed to accrual 07/27/2018) 2. Epithelial tumors of major salivary glands (closed to accrual 03/20/2018) 3. Salivary gland type tumors of head and neck, lip, esophagus, stomach, trachea and lung, breast and other location (closed to accrual) 4. Undifferentiated carcinoma of gastrointestinal (GI) tract 5. Adenocarcinoma with variants of small intestine (closed to accrual 05/10/2018) 6. Squamous cell carcinoma with variants of GI tract (stomach small intestine, colon, rectum, pancre