Bradley played high school baseball and football in Macomb, Illinois for the Macomb High Bombers. Due to his success there, the Macomb High School baseball field was later dedicated in his name. Also a talented football player, he played college football at the University of Missouri in Columbia, Missouri and was their starting quarterback from 1978 through 1980.
One of the most decorated athletes in MU annals, Bradley lettered in football at MU from 1977–81, and in baseball in 1979-80-81. Bradley quarterbacked the Tigers to three bowl games. He was a three-time Big Eight Conference "Offensive Player of the Year" and set the conference total offense record at 6,459 yards which stood for 10 years. In baseball, he starred as an outfielder on MU teams that won the Big Eight championship in 1980, and went to the NCAA Tournament in 1980 and '81.
He was drafted by the Seattle Mariners in the third round of the 1981 amateur draft and made his Major League debut on September 2, 1983, as a pinch hitter against the New York Yankees.
Bradley became Seattle's regular left fielder in 1984, batting .301 in 124 games, but did not show any power, hitting no home runs and only three in his career in the minors to that point. In 1985, he hit .300 in 159 games, adding a surprising 26 home runs, and was selected to the American League All-Star team. Bradley was a productive player in Seattle, never hitting below .297 in four full seasons while also stealing 107 bases. On April 13, 1985, with two outs in the ninth inning, Bradley hit a walk-off grand slam home run to win by one run, becoming the third American League player to do so (ninth player in the majors). On April 29, 1986, Bradley was Roger Clemens' 20th and final strikeout as the pitcher set a major league record for strikeouts in a game. In December 1987, the Mariners traded Bradley and Tim Fortugno to the Philadelphia Phillies in exchange for Mike Jackson, Glenn Wilson, and minor leaguer Dave Brundage.
Bradley hit a respectable .264 in his only season with the Phillies. While with Philly, Bradley was hit by a pitch 16 times during the season which set a Phillies team record. Almost one year to the day since arriving from the Mariners, the Phillies, desperately in need of pitching help, dealt Bradley to the Baltimore Orioles for Gordon Dillard and Ken Howell.
Back in the more familiar American League, Bradley's batting average rose to .277 in his first season in Baltimore. In mid-season 1990, he was traded to the Chicago White Sox for Ron Kittle. His final major league appearance came on September 29, 1990, as he drew two walks and scored a run in a 5-2 White Sox win over the Seattle Mariners.
Shortly after retirement, he was hired as the baseball coach at Westminster College in Fulton, Missouri. He also taught classes there, including upper level classes on sports history.
In September 2009, Bradley was named as a volunteer assistant coach of the University of Missouri softball team for the 2009-10 season. He is currently a Special Assistant to the Executive Director for the Major League Baseball Players Association.