Bogut is the son of Croatian immigrants to Australia. As a 15-year-old, he was cut from the Victoria junior state representative team. That experience apparently drove him to improve; he later went on to attend the Australian Institute of Sport. In his last season at AIS, he averaged 22 points and 14.5 rebounds. He went on to lead the Australian team that won the 2003 FIBA Junior World Championships, and was named the tournament MVP.

As a freshman at Utah in 2003-04, he barely missed averaging a double-double for the season with 12.5 points and 9.9 rebounds, and was named Freshman of the Year in the Mountain West Conference. During the summer, he was a starter on the Australian Olympic team, averaging 14.8 points, 8.8 rebounds, and 1.2 blocked shots and shooting 58.0% from the field.

After his solid though unspectacular freshman season, he did not even receive honorable mention on the preseason All-America team for the 2004-05 season. However, he had a true breakout season as a sophomore. Bogut averaged 20.4 points, 12.2 rebounds (second in Division I), 2.3 assists, and 1.8 blocks, and shot 62.0% from the field (eighth in Division I). He also led Division I in double-doubles, with 26.

After the 2004-05 regular season, Bogut was the leading vote-getter on the AP All-America team and won most major national individual awards, earning Player of the Year honors from the Associated Press and ESPN.com, plus the Naismith and Wooden Awards. Bogut is the first non-U.S. player ever to win either award. (Patrick Ewing, who won the Naismith in 1985, was born in Jamaica, but was naturalized by the time of his award, and 1997 Naismith and Wooden winner Tim Duncan is a native of the U.S. Virgin Islands, making him a U.S. citizen by birth.)

During an interview in the hours leading up to the draft, Bogut angered some sports commentators in Australia by referring to himself as the greatest Australian basketball player, while making derogatory comments about former champions and NBA players, including Andrew Gaze and Luc Longley. He apologized for the comments later in the day, sending a personal email to Gaze. He was also warned by Luc Longley and other people of wisdom to not allow his "big-head" to bloat too much.

Whilst many Australian players waved off the comments, former NBA player and current Australian captain Chris Anstey did not. Speaking on Australian national television, the 30-year-old Anstey challenged the 20-year-old Bogut to a 1-on-1 showdown "anywhere, anytime, and I would beat him."

Bogut's checkered infamy continued into his 2005 NBA summer league play. With many rank and file players looking to prove themselves against Bogut, he struggled for consistent form and was often outmuscled and outplayed. Most notably he was thrown out of a game against the Indiana Pacers after attempting to remonstrate against Pacers center John Edwards (Edwards was also ejected). Bogut ended his summer league play averaging 13.2 points per game and 10 rebounds.

He has now headed back to Australia and plans to add weight for the upcoming season.