Additional Study Information: This study is for adult subjects with head and neck cancer (oral cavity, oropharynx middle part of the throat, hypopharynx bottom part of the throat, or larynx part of throat between the base of the tongue and the windpipe) that has either returned after treatment (recurrent) or is newly diagnosed. If newly diagnosed, the cancer must be metastatic, meaning it has spread to another area of the body. This study includes the treatment of subjects with two study drugs that uses a bodys own immune system to fight cancer. The study drugs administered in this study, MEDI4736 and tremelimumab, are laboratory-created antibodies. Antibodies are proteins produced by the human bodys immune system that bind to other proteins and molecules. The main purpose of this study is to determine the doses and dosing schedule of both MEDI4736 and tremelimumab that are safe and tolerable when given together in subjects with certain types of head and neck cancer.This study will also evaluate the ability of MEDI4736 and tremelimumab or tremelimumab alone to shrink the tumor(s).