It may have been the occasion where one of his brothers was jailed for attempted robbery after shooting a pregnant woman; she survived, but the foetus did not. Perhaps it was when his best friend was shot dead over a gambling debt, or maybe it was when he was arrested in college for stealing hub-caps from a car.
Alternatively, it may have been the terror-filled night when his father returned home drunk at 2am, woke all six of his sons, lined them up against a wall and shot a bullet over each of their heads. It was done, said Joe Davis, to toughen up his offspring. "I brought you into this world, and I can take you out," he said. "I wasn't scared," said Terrell. "I knew he wasn't going to kill us. He loved us too much."
Whatever the intention, Davis is undeniably tough nowadays, an essential quality for a running back who pays a demanding physical price every time he carries the ball. In four years with the Broncos the 26-year-old from San Diego has proved virtually unstoppable, and if the Atlanta Falcons fail to devise a way of stopping him tonight, Super Bowl XXXIII will be a one-sided affair.
Some players are marked for greatness during their college years, but not Davis. An undistinguished career at the University of Georgia, where he was troubled with a persistent groin injury, meant that he was a lowly sixth-round selection in the 1995 draft. Some 195 players, including 20 running backs, were taken before him. A long shot simply to make the Broncos roster, Davis caught the eye in training camp and when the season opened six weeks later he was in the starting line-up.
In just four seasons he has claimed 59 Denver records. He became only the fourth player in history to gain 2,000 yards in a season when he rushed for 2,008 this year, and it is only a matter of time before he breaks Eric Dickerson's single-season mark of 2,105. His 6,413 career yards are the most by anyone after four years, while his average of 105.1 yards per game is the most productive of all time. He is already drawing comparisons with the greats, Jim Brown, Walter Payton and Barry Saunders: ultimately he may prove to be better than all of them.
"I just don't see any weaknesses," said the Atlanta linebacker Jessi Tuggle, one of the players attempting to halt his progress tonight. "I look at tapes to see if he slows down or if he is slipping up. You try to find a weakness but there's not many. He gives you everything from the first quarter to the fourth quarter."
Of greater concern to the Falcons is the knowledge that Davis is at his best on big occasions. In last year's Super Bowl, his 157 yards and three touchdowns were the difference in the Broncos' 35-31 upset win over the Green Bay Packers. The achievement was even more remarkable considering that Davis missed the second quarter, suffering from one of the migraines which have blighted him since childhood. In this year's play-off games, he pounded the Miami Dolphins for 199 yards, following up with a further 164 against the New York Jets. The bigger the game, the more he performs. "That's the time you have to play well, because if you lose in the play-offs, you're going home," he said. "I recognise that when I am on the field I don't have a second chance, so I concentrate on playing the best game I can."
Team-mates and rivals have acknowledged the unique combination of talents that have helped produce one of the best running backs ever. "The thing I really like about Terrell is that every year his open-field running is more and more impressive," said his Atlanta counterpart Jamal Anderson. "That's difficult for a running back. His cutting and his vision, those things are improving constantly and it's scary."
"He has everything you look for in a running back," said Denver's head coach, Mike Shanahan. "He's the only guy I have ever been around who doesn't have a weakness. He's extremely smart and dedicated. He's got elusiveness, and the speed to make the long run. He can pass protect and he can block. He is the complete back. Plus, he's a great team player: he puts the team in front of any individual goal."
Last week Davis was officially named the League's Most Valuable Player for 1998, a foregone conclusion given his contributions to a team who won 14 of 16 regular season matches and are widely expected to win the Super Bowl. After an uncertain past, Davis looks to an exciting future. He says he wants another four or five titles, and that when he retires he wants to be acknowledged as the best ever. "I would like to think I am the greatest back ever," he said. "When I leave this game, I want to leave my signature on the game."
Another match-winning performance tonight, and that signature should have quite a flourish.
Case for the offense and defense
Experience: Alien territory for the Dirty Birds - 60-1 at the start of the season. Never been beyond the last eight in their 33-year history.
Coaching: Dan Reeves will be involved in a record ninth Super Bowl. Took Denver to three in the 1980s, but lost the lot.
Strengths: Impressive running attack with Jamal Anderson. Reliable quarterback in Chris Chandler, plus two solid receivers. Defensively very tight, the Falcons rely on containment and pouncing on errors.
Weaknesses: Inexperienced offensive line. Play in dome on astroturf. Super Bowl is outdoors on grass. No Dome team has won the big one.
Key player: Linebacker Jessie Tuggle, 12-year veteran, who must have the game of his life to slow down Denver's prodigious running attack.
Unsung hero: Kicker Morten Andersen. The NFL's only Dane showed cool head in kicking game-tying extra point, then overtime field goal, to defeat Minnesota. Perfect temperament for a pressure situation.
Can they win? They have the potential for an upset, but unless they find a way to stop Terrell Davis, it looks a tall order. The longer it stays close, the more they will fancy their chances.
Experience: This is the franchise's sixth Super Bowl, but their only victory came last year against Green Bay.
Coaching: Mike Shanahan won a Super Bowl as assistant with San Francisco, and as head man last year. Outstanding offensive tactician.
Stengths: The best running game in the league, the premier running back, quarterback of legendary status, formidable offensive line, punishing run stoppers on defense.
Weaknesses: Defensive secondary small and ageing, might be vulnerable to the pass. Whole team will need to guard against complacency.
Key player: Quarterback John Elway. If Terrell Davis should falter, the 16-year veteran Elway, in quite possibly his last game, will rise to the occasion.
Unsung hero: Defensive tackle Keith Traylor. Once discarded by the Broncos, revitalised his career with Barcelona Dragons in NFL Europe, and has become one of the game's arch run stoppers.
Can they lose? If Davis finds his rhythm, the Broncos will retain their title. If they manage to build a 14-point lead, they will not be caught.