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, pancreas) (closed to accrual 10/17/2018) 7. Fibromixoma and low grade mucinous adenocarcinoma (pseudomixoma peritonei) of the appendix and ovary (closed to accrual 03/20/2018) 8. Rare pancreatic tumors including acinar cell carcinoma, mucinous cystadenocarcinoma or serous cystadenocarcinoma. Pancreatic adenocarcinoma is not eligible (closed to accrual) 9. Intrahepatic cholangiocarcinoma (closed to accrual 03/20/2018) 10. Extrahepatic cholangiocarcinoma and bile duct tumors (closed to accrual 03/20/2018) 11. Sarcomatoid carcinoma of lung 12. Bronchoalveolar carcinoma lung. This condition is now also referred to as adenocarcinoma in situ, minimally invasive adenocarcinoma, lepidic predominant adenocarcinoma, or invasive mucinous adenocarcinoma 13. Non-epithelial tumors of the ovary: A) Germ cell tumor of ovary B) Mullerian mixed tumor and adenosarcoma (closed to accrual 03/30/2018) 14. Trophoblastic tumor: A) Choriocarcinoma (closed to accrual) 15. Transitional cell carcinoma other than that of the renal, pelvis, ureter, or bladder (closed to accrual) 16. Cell tumor of the testes and extragonadal germ tumors: A) Seminoma and testicular sex cord cancer B) Non seminomatous tumor C) Teratoma with malignant transformation (closed to accrual) 17. Epithelial tumors of penis - squamous adenocarcinoma cell carcinoma with variants of penis (closed to accrual) 18. Squamous cell carcinoma variants of the genitourinary (GU) system 19. Spindle cell carcinoma of kidney, pelvis, ureter 20. Adenocarcinoma with variants of GU system (excluding prostate cancer) (closed to accrual 07/27/2018) 21. Odontogenic malignant tumors 22. Pancreatic neuroendocrine tumor (PNET) (formerly named: Endocrine carcinoma of pancreas and digestive tract.) (closed to accrual) 23. Neuroendocrine carcinoma including carcinoid of the lung (closed to accrual 12/19/2017) 24. Pheochromocytoma, malignant (closed to accrual) 25. Paraganglioma (closed to accrual 11/29/2018) 26. Carcinomas of pituitary gland, thyroid gland parathyroid gland and adrenal cortex (closed to accrual) 27. Desmoid tumors 28. Peripheral nerve sheath tumors and NF1-related tumors (closed to accrual 09/19/2018) 29. Malignant giant cell tumors 30. Chordoma (closed to accrual 11/29/2018) 31. Adrenal cortical tumors (closed to accrual 06/27/2018) 32. Tumor of unknown primary (Cancer of Unknown Primary; CuP) (closed to accrual 12/22/2017) 33. Not Otherwise Categorized (NOC) Rare Tumors \[To obtain permission to enroll in the NOC cohort, contact: S1609SC@swog.org\] (closed to accrual 03/15/2019) 34. Adenoid cystic carcinoma (closed to accrual 02/06/2018) 35. Vulvar cancer (closed to accrual) 36. MetaPLASTIC carcinoma (of the breast) (closed to accrual) 37. Gastrointestinal stromal tumor (GIST) (closed to accrual 09/26/2018) 38. Perivascular epithelioid cell tumor (PEComa) 39. Apocrine tumors/extramammary Paget's disease (closed to accrual) 40. Peritoneal mesothelioma 41. Basal cell carcinoma (temporarily closed to accrual 04/29/2020) 42. Clear cell cervical cancer 43. Esthenioneuroblastoma (closed to accrual) 44. Endometrial carcinosarcoma (malignant mixed Mullerian tumors) (closed to accrual) 45. Clear cell endometrial cancer 46. Clear cell ovarian cancer (closed to accrual) 47. Gestational trophoblastic disease (GTD) 48. Gallbladder cancer 49. Small cell carcinoma of the ovary, hypercalcemic type 50. PD-L1 amplified tumors 51. Angiosarcoma 52. High-grade neuroendocrine carcinoma (pancreatic neuroendocrine tumor \[PNET\] should be enrolled in Cohort 22; prostatic neuroendocrine carcinomas should be enrolled into Cohort 53). Small cell lung cancer is not eligible (closed to accrual) 53. Treatment-emergent small-cell neuroendocrine prostate cancer (t-SCNC)
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Overall response rate (ORR)
Timeframe: Up to 10 years