Adams died at Self Regional Hospital after going into cardiac arrest about an hour earlier 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 often can lead to a heart attack, Kelley-Clark said. She said relatives were unaware of any medical condition.
Toxicology tests are being run by the State Law Enforcement Division, though drug use is not suspected. The results probably will not be available for at least two months, Kelley-Clark said.
Adams, 6-5 and 258 pounds, spent three seasons in the NFL, including two full years with the Buccaneers before splitting this season between the Bucs and Bears. "He was a true team player and a positive influence to everyone he met," Bucs coach Raheem Morris said.
The Bucs drafted Adams fourth overall in 2007, but he was not able to live up to expectations that he would revive their once-feared pass rush. He had only 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," Bucs cornerback Ronde Barber said. "He had so much potential that had yet to be achieved."
Bucs defensive tackle Chris Hovan said he took Adams under his wing when he came to Tampa Bay.
"I considered him my little brother,'' Hovan said, "and that's how I will always remember him. This is all so unreal, and it hasn't really hit me yet."
Morris said at the start of training camp that Adams would be considered a "bust" if he didn't reach double-digits in sacks. Adams fell short of the benchmark, although he welcomed the challenge.
"In football you need that," Adams said in August. "Players tend to get in their own element and do things that they want to do. They need to be called out sometimes. He's the coach. Whatever he says goes."
Morris criticized Adams after lackluster performances in the first three games. With the Bears, Adams played brief stints on defense. He made five tackles.
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."