After graduating from high school in Frederick, Oklahoma in 1941, Bobby entered University of Tulsa to play football for coach Henry Frnka.
With the United States involved in World War II, Bobby's patriotism, along with his desire to become a pilot and play football, led him to West Point.
The 1944 Army team had a 9–0 record, two future Heisman Trophy winners (Doc Blanchard & Glenn Davis), and defeated their opponents by a combined score of 504–35 en route to winning the Associated Press National Championship.
In 1952, Earl Blaik brought Dobbs back to West Point to serve as an assistant football coach.
His 1967 UTEP team led the nation in passing and scoring that season, losing its two games by a total of three points.
In April 1966, Dobbs turned down an offer to succeed Paul Dietzel as Army Black Knights coach due to his wife, Joanne's illness that required her to stay in the warm climate in El Paso.
While at Texas Western, he sent many players into the pro ranks, with quite a few going, including Fred Carr, Billy Stevens, Ron Jones, and Leon Harden to Green Bay to play for Vince Lombardi.
Bobby's failing health turned out to be Alzheimer's disease, and he died on April 2, 1986, in a nursing home in Altus, Oklahoma.