He died at Self Regional Hospital after going into cardiac arrest about an hour before at his family's home in Greenwood, said Marcia Kelley-Clark, chief deputy coroner for Greenwood County.
An autopsy showed an enlarged heart, a condition that can often lead to a heart attack, Kelley-Clark added. She said relatives were unaware of any medical condition.
Adams, 6-foot-5 and 258 pounds, spent three seasons in the NFL, two with the Buccaneers and part of this season with the Bears.
"He was a true team player and a positive influence to everyone he met," Buccaneers coach Raheem Morris said.
Adams was selected fourth overall in the 2007 draft by Tampa Bay. He had not been able to live up to expectations that he would revive the Buccaneers' once-feared pass rush, and had just 17 tackles and one sack in 15 games -- 10 with Chicago -- this season.
He was traded to the Bears in October for a second-round pick in the 2010 draft.
"Gaines was a quiet, humble kid and is far too young to be gone," Buccaneers cornerback Ronde Barber said. "He had so much potential that had yet to be achieved."
Buccaneers defensive tackle Chris Hovan said he took Gaines under his wing when he came to Tampa Bay.
"I considered him my little brother and that's how I will always remember him," he said. "This is all so unreal and it hasn't really hit me yet."
Clemson fans recall Adams' performance in a 2006 victory over Wake Forest in which he broke up a field goal try and returned it for a touchdown.
Tommy Bowden, Adams' coach at Clemson, was jolted by news of his death.
"I just couldn't believe it was Gaines," he said. "I will always remember the smile he had on his face and I will always remember his patience."