Curry accepted an athletic scholarship to attend the University of Alabama, where he played for coach Bill Curry and coach Gene Stallings' Alabama Crimson Tide football teams from 1989 to 1992. As a senior in 1992, Curry was recognized as a consensus first-team All-American and the United Press International Lineman of the Year. He was a member of the Crimson Tide squad that won the first SEC Championship Game in 1992, and a consensus national championship by defeating the Miami Hurricanes 34–13 in the Sugar Bowl. Curry and fellow defensive end John Copeland were known as the "bookends" during their time at Alabama, and were key players in the Crimson Tide's No. 1 rated defense in 1992.
The Tampa Bay Buccaneers selected Curry in the first round, with the sixth overall pick, of the 1993 NFL Draft. He gained notoriety for happily hugging the NFL Commissioner, Paul Tagliabue, after he was picked by the Buccaneers in the draft. He played for the Buccaneers from 1993 to 1997. He signed a free agent deal with the Green Bay Packers in 1998, but was released during the preseason. Curry played for the Jacksonville Jaguars from 1998 to 1999. In seven NFL seasons, Curry played in 75 regular season games, started 44 of them, and contributed 95 tackles, 12.5 quarterback sacks, and seven forced fumbles.