This question raises a couple of topics:
What is the minimum amount of time per move a player needs in order to avoid blunders, displacing the pieces, fingerfehlers, etc.
When the time handicap is considerable, how do you prevent the stronger player from profiting from the weaker player's time?
Is there a rating (or rating gap) beyond which additional time (or a larger handicap) makes no appreciable difference to the outcome?
Q1. Analyzing the blitz games of GM's, I've found (unsurprisingly) that they're significantly more vulnerable to blundering at 3 minutes a game. This is true for both the eventual winner and the loser, suggesting that neither morale nor satisfaction with one's position is solely responsible for this tendency. It also occurs (typically) multiple times per game.
I have a collection of games played on ICC at different time controls, and I noticed that the same players use both sets of time controls. This might be a valuable testing ground for the effect of time controls on game performance (at least, as reduced to blunder occurrence rates). But, no conclusions yet. I would expect to find the same is true; although it likely varies by player, there is probably some minimum time control at which blunders begin to appear with frequency and/or magnitude that dominate the play of the game.
Q2. I might suggest that the stronger player be distracted. On ICC, on Wednesdays (currently), an International Master with the handle Voja plays a simultaneous display against his opponents. Nominally, their time control is 3 minutes + 3 seconds delay, while his is 1 minute plus 1 minute delay. As in most simuls, he makes a move and goes on the next board/game. So, he is not able to think about Game 20 while selecting moves for Games 21-30 and then Games 1-19. He only gets to think once he returns, and that's when his clock starts again. His opponents, however, have the clock time to make their move, but then they get to try to anticipate his move and think ahead until he returns to the board. Lower rated players (1700 ELO and up) are often able to beat him with this kind of time handicap. (Of course, once the number of opponents has been whittled down to a few and he's returning more quickly to each board, the 3/3 control begins to take hold, and there's no borrowed time anymore).
Q3. I would suspect that below a certain rating, say 1500 ELO or so, certain critical aspects that may develop in a position are not visible to the player. So, no amount of additional time will help them find the right move. Potential discovered attack, decoy, and trapped piece tactics that are one or two moves away seem to fall into this category, in my experience, but that's subjective (so far).
I think the evidence that some 1700 players can beat an IM at blitz on ICC when the time controls are substantially weighted in their favor answers the question about gap; below 1700, they don't seem to be able to do this. (The ELO gap involved is around 600 ELO.) In that situation, it may not make a difference for them to have more time.