Over on his website, Charles Staley wrote an excellent article on training to failure. That's when you push an exercise until the point you cannot continue to perform it (either with good form - technical failure, or at all - muscular failure).
Training to Failure
The article goes into the background on failure training, as well as the possible downsides for doing so. It's pretty in-depth but I think it's useful for anyone still not convinced there is a way to train that isn't "until I can't do any more reps."
He also makes a very clear statement I think is worth remember for all trainees, not matter how you train:
"Progress is a function of gradually increasing your training load over time—not how "trashed" you feel after a workout." - Charles Staley
You can substitute "training load" with "skill level" for sports-specific training, too. If you're training to run faster, a new record in your run time is a better indicator of progress than "I ran myself into the dirt last time!" If you're competitive athlete, your win-loss record, batting average, shooting percentage, whatever, is a better indication than tiredness or perceived exertion. Train enough to get the best improvement you can, and no harder!