Bright earned a bachelor's degree in linguistics in 1949 and a doctorate in the same field in 1955, both from the University of California, Berkeley. He was a professor of linguistics and anthropology at UCLA from 1959 to 1988. He then moved to the University of Colorado at Boulder, where he remained on the faculty until his death.
Bright was an authority on the native languages and cultures of California, and was especially known for his work on Karuk, a Native American language from northwestern California. He was made an honorary member of the Karuk tribe -- the first outsider to be so honored -- in recognition of his efforts to document and preserve their language which led to its revival. Bright was also known for his research on the native American languages Nahuatl, Kaqchikel, Luiseño, Ute, Wishram, and Yurok, and the South Asian languages Lushai, Kannada, Tamil, and Tulu.
Bright was editor of Language, the journal of the Linguistic Society of America, from 1966 to 1988 and of Language in Society from 1993 to 1999. He was the founding editor of Written Language and Literacy, which he edited from 1997 until 2003. He served as President of the Linguistic Society of America in 1989.