In the case of The Teeth, he and his agent threatened he'd play baseball rather than play for the 1983 Colts.
The Teeth also told the Colts he wanted to play on the West Coast or for the Cowboys or Dolphins. The Colts kept trying to make trades, but no trade happened before the draft, probably because the other teams knew the Colts had no leverage and were getting into a weaker position each day.
The Colts informed The Teeth just before the draft that since they hadn't gotten a trade done, they would have to pick him. Once the draft started, the Colts picked The Teeth immediately. They had 15 minutes, but put in their choice right at the beginning. The Teeth said publicly that he was going to play baseball. The Teeth's father said The Teeth would never play for Irsay or Colts head coach Frank Kush.
Irsay had been negotiating with the Broncos since before the draft. The Colts were interested in offensive lineman Chris Hinton, whom the Broncos had taken with the fourth pick. In May of '83, the Colts traded The Teeth to the Broncos for Hinton, a backup QB, and a first-round pick in the next year's draft.
Hinton was drafted to play guard and played in all 16 games for the Colts as a left guard, starting 15 of them, and later moved successfully to left tackle for the Colts, then later to right tackle and right guard for the Falcons and Vikings. He made the Pro Bowl seven times - six for the Colts and once later for the Falcons - in his 13 seasons in the league. He was second-team All-Pro three times, all as a tackle for the Colts, and first-team All-Pro twice, once as tackle for the Colts and once as a guard for the Falcons.
The Broncos' first-round draft pick turned out to be pick number 19 in the 1984 draft, which the Colts turned into offensive lineman Ron Solt. Solt started 16, 15, and 16 games in 1984-86, all as a RG. In 1987, Solt played in 12 games, starting all of them. He made the Pro Bowl and was second-team All-Pro as a guard. He then got into a contract dispute with the Colts and held out for the first month of the 1988 season, and then was traded to the Eagles for the Eagles' 1989 first-round pick and 1990 fourth-round pick.
The 1989 first-rounder became Andre Rison, who played his rookie season for the Colts, then was traded, along with Hinton, the Colts' 1991 first-round draft pick, and the Colts' 1990 fifth-round pick, to Atlanta for Atlanta's #1 pick in the 1990 draft. Y'know, so the Colts could draft Jeff George. Two picks later, the Seahawks got Tez.
The Falcons used the Colts' 1991 first-round pick to draft WR Mike Pritchard, who played nine seasons in the NFL - for the Falcons in 1991-93, the Broncos in '94-'95, and the Seahawks in '96-'99.
The 1990 fourth-rounder became offensive lineman Rick Cunningham, who appeared in two games for the Colts, but not as a starter, in 1990 before bouncing around the league, playing in 80 games for four teams over eight seasons.
What happened to all the pieces of what the Colts managed to get from the Broncos for The Teeth is interesting. Given the situation, it's surprising they were able to get as much as they did for him.