I recently asked a question about why the Jedi might not train Force users, but as I was asking that, I realized that there was a second question...
What happens to those who are denied training? Presumably Anakin wasn't the only Force user where the question of whether or not to train them was raised.
As mentioned in the answers on my other question, there's a possibility that the Sith might draw these over to the Dark Side... so do the Jedi put them on ice? Do they simply monitor these individuals?
It seems like there would be too few Jedi to monitor everyone who's shown some kind of ability with the Force...