Bedford was born in Beaumont, Texas. He played high school football at Hebert High School for his legendary father Leon Bedford where he was an all-District player on the first all-black high school to win a University Interscholastic League state title in Texas.
He played college football at the University of Texas at Austin where he was a four-year letterman and starter at cornerback as well as the defensive captain. He was a two-time All-Southwest Conference second team selection. He played in two Cotton Bowls and two Sun Bowls. He set a then-Longhorn season record for pass breakups with 22 in 1981 and is currently in the top ten on UT's career pass breakup list (47). At the end of his senior year, he was named a Defensive Valuable Player in the 1982 Senior Bowl All-Star Game. He returned to Texas to receive his diploma in 1984.