Bradley was born and raised in Crystal City, Missouri. He excelled at basketball from an early age and was offered 75 college scholarships but declined them all to attend Princeton University. He won an Olympic Gold Medal as a member of the 1964 Olympic Basketball Team and was the NCAA Player of the Year in 1965 when Princeton finished third in the NCAA Tournament. After graduating in 1965, he attended Oxford on a Rhodes Scholarship, delaying a decision for two years on whether or not to play in the NBA.
While at Oxford, Bradley played one season of professional basketball in Europe and eventually decided to join the New York Knicks in the 1967–68 season, after serving six months in the Air Force Reserve He spent his entire ten-year professional basketball career playing for the Knicks, winning NBA titles in 1970 and 1973.
After the NBA, Bradley has spent much of his career in politics.