Bedford attended C. K. McClatchy High School. He received Bee All-Metro honors at quarterback as a senior. In basketball, he set the national high school record for season field-goal percentage as a senior.[1]
He accepted a football scholarship from the University of California. He was considered a dual-threat at quarterback, with the ability to both pass and run. As a freshman, he was a backup behind Gale Gilbert.
As a sophomore, he was a mostly a backup behind Kevin Brown, but still had a chance to start a few games. In the season finale against Stanford University, he replaced an ineffective Brown with the team trailing 0-24 at halftime. Bedford led a remarkable third quarter comeback that put the Golden Bears in a position to win the game, but sprained his ankle while celebrating a touchdown, forcing him to miss most of the fourth quarter and having to watch kicker Leland Rix miss a late 30-yard field goal in a 22-24 loss.[2] During the season, he tallied 46-of-103 completions for 627 passing yards, 157 rushing yards, 2 passing touchdowns, 6 rushing touchdowns and 7 interceptions.
As a junior, he was named the starting quarterback. In the season opener, he completed 4-of-16 attempts for 32 yards in a 15-21 loss against Boston College.[3] He was replaced 4 games into the season with true freshman Troy Taylor. Taylor suffered a broken in the tenth game against USC. Brown would be the player chosen to finish the game, but also start in the season finale against Stanford University, engineering a 17-11 upset win.[4] Bedford finished with 42-of-93 completions for 488 yards, 3 passing touchdowns and 5 interceptions.
As a senior, he was converted into a wide receiver, leading the team with 39 receptions for 515 yards and 4 receiving touchdowns, while also having 7 carries for 53 yards (7.6-yard average).
On September 19, 1990, he was signed by the BC Lions of the Canadian Football League.[11] He was limited with a right knee injury, registering only 4 receptions for 54 yards.