Bradley was born in Landstuhl, West Germany, as his family was stationed at the U.S. military base medical facility, and grew up in Castle Dale, Utah.

He was one of the most successful basketball players in Utah high school history, finishing with career averages of 20.3 points, 11.5 rebounds, and 5.4 blocked shots per game.

He turned in a strong performance at the McDonald's All-American Game (12 points, 10 rebounds, and 6 blocks), earning the Most Valuable Player award for the West team.

[9] Bradley was heavily recruited by many colleges from across the United States, but ultimately decided to play at Brigham Young University (BYU) in Provo, Utah.

But his main contribution to the team was shot-blocking: he led the entire nation in total blocks (177) and average (5.2 per game)—setting an NCAA freshman record in both categories.

[10] In one game against Eastern Kentucky (EKU), Bradley blocked 14 shots, tying the NCAA single-game record set by Navy's David Robinson in 1986.

He was also named Honorable Mention All-American by the Associated Press (AP)[14] Bradley took a two-year leave of absence from BYU from 1991 to 1993 to serve as a full-time missionary for the LDS Church.

Critics argued that he would struggle in the NBA due to his thin build – listed at just 235 pounds (107 kg), he was described by Ric Bucher of The Miami Herald as "rail-thin," and Lee Rose, player personnel director for the Milwaukee Bucks at the time, said, "Size-wise, he'll get pushed around a little bit,"[15] – and lack of experience (just one season of college basketball, followed by a two-year hiatus from the sport due to his church mission).

The team acquired legendary big man Moses Malone to serve as a mentor for Bradley, hoping to develop his skills in the low post.

[17] Also, the franchise hired strength and conditioning consultant Pat Croce and registered dietitian Jeanie Subach, as well as assistant coach Jeff Ruland, to add bulk to Bradley's skinny frame.

Bradley gave Philly fans a glimpse of his potential in a game against the Los Angeles Clippers in November, posting career highs of 28 points and 22 rebounds in a 97–83 victory.

He posted a career-high 32 points and added 15 rebounds against Dallas early in the month,[23] and set another career high a week later with 10 blocked shots against the Phoenix Suns.

In April, Bradley posted back-to-back triple-doubles (against Boston and Toronto); in the process, he became the first player in NBA history to record consecutive games of double-figure blocked shots twice in a single season.

[10] The previous players to record a 20–20–10 game with points, rebounds, and blocks were Kareem Abdul-Jabbar, Elvin Hayes, Hakeem Olajuwon, and Shaquille O'Neal.

Over that span, his playing time slowly declined, mainly due to some injuries and constant juggling of the team's lineups and rotations by coach Don Nelson.

The Mavericks emerged as a perennial playoff contender, mostly due to the rise of star players such as Dirk Nowitzki and Steve Nash, but Bradley's defense made a significant contribution.

[27] Bradley has competed internationally for Germany; he and Dallas teammate Dirk Nowitzki were members of the German national basketball team that finished in fourth place at the EuroBasket 2001 in Turkey.

[24][33] Bradley has also participated in the "Basketball Without Borders" program with other NBA players like Dikembe Mutombo, DeSagana Diop and Malik Rose and found his experiences with children in Africa uplifting.

[37] On January 20, 2021, Bradley was struck from behind by a motorist while riding his bicycle near his home in St. George, Utah, which resulted in a traumatic spinal cord injury in his neck that left him paralyzed.

[38] Bradley was in the process of passing a car parked on the shoulder of the road when he drifted in front of a minivan who bumped him from behind; the driver of that vehicle was not charged with a crime.

[37] Bradley had a role in the film Space Jam in 1996,[39] depicted as one of the NBA stars who lose their talent alongside Muggsy Bogues, Larry Johnson, Charles Barkley, and Patrick Ewing.

[39] In 2011, he appeared in a music video tribute to Jimmer Fredette, along with former BYU football star Chad Lewis and legendary coach LaVell Edwards.