Then along came coach Bruce Pearl, who has the Vols in the NCAA regional semifinals in his second season.
Tennessee (24-10) beat Virginia to reach the round of 16 for the first time since 2000, and plays Ohio State, the top seed in the South Regional, on Thursday night in San Antonio.
Tennessee has never advanced to the regional final.
Bradshaw is the Vols' lone senior, first enrolling at the university in 2004, the second-to-last year for coach Buzz Peterson. That season the Vols lost at George Mason in the opening round of the NIT, and they failed to reach either the NIT or NCAA in 2005.
"The first couple of years, it's hard to take anything good out of it, but all of sudden I'm in this position my last year. I feel proud to be one of the guys that went through the struggle first to enjoy the success. It's made it that much sweeter," Bradshaw said Monday after a light practice back on campus.
Bradshaw is one of the best examples of how times have changed on Rocky Top since Pearl arrived after leading Wisconsin-Milwaukee on a run to the round of 16 in 2005.
Bradshaw is a backup point guard, moved to forward. Even though Bradshaw is usually much smaller (6-foot-4) than the players he guards, Pearl wanted to find some way to get Bradshaw on the court.
Now he's a fan favorite for his gutsy plays that don't always show up in the box score. Against Virginia, Bradshaw had to inbound the ball three times in the final seconds to star shooter Chris Lofton, who was heavily defended each time. Lofton went 6-of-6 at the foul line down the stretch to seal the 77-74 victory Sunday.
The Vols have historically struggled in the NCAA tournament. In 15 appearances, this is only the second time they have won two games in a row. The Vols' previous trip to the regional semifinals was in 1981 when they had a first-round bye in the field of 48.
Last year Tennessee won in the first round but lost to Wichita State in the second round, failing to capitalize on a No. 2 seed.
Pearl wanted to do better this season, and now he wants more.
"If we're going to be a top-20 program, you've got to play in the Sweet 16," Pearl said.
"I don't know exactly when we needed to get to the Sweet 16; obviously within a timeframe the fans kind of would like their coaches and their teams to get there. I believe we're a top-20 program, but the only way I can say it and not be laughed at is if you do (it) with some consistency."
Tennessee has played six of the teams still playing in the tournament - including Southeastern Conference East division opponents Florida and Vanderbilt twice each.
That list includes top-seeded Ohio State, which beat Tennessee 68-66 on Jan. 13.
The Vols noticed the Buckeyes lurking in their bracket the day it was announced, and they were already thinking about it after the second round.
"It was fun to get off the plane and walk through the airport and have the Knoxville faithful cheering for us," Bradshaw said. "There's no doubt it was a great moment for all of us, but we're still looking forward to hopefully advancing."