Born in Del Rio, Texas, Bunton was a private in the United States Army, 76th Infantry Division, during World War II, from 1943 to 1946.

He received a Bachelor of Arts degree from the University of Texas at Austin in 1947 and a Juris Doctor from the University of Texas School of Law in 1950.

[1] On October 11, 1979, Bunton was nominated by President Jimmy Carter to a seat on the United States District Court for the Western District of Texas vacated by Judge John H. Wood Jr. Bunton was confirmed by the United States Senate on November 26, 1979, and received his commission on November 27, 1979.

He served as Chief Judge from 1987 to 1992, assuming senior status on December 1, 1992.

[1] Bunton took inactive senior status in May 2000 for health reasons due to a diagnosis of bladder cancer.