I sometimes ride brakeless and seldom use the brake on my daily commuter bike, though I had the peace of mind knowing it's there if I need it. I can effectively skid and skip. I think I am a good rider.
However, having said that, you can't really stop SHORT, as in on a dime on a brakeless fixie. I'm talking about if you have any reasonable amount of momentum. Obviously, if you are putting around, you can stop on a dime.
But, as others mentioned, its not about stopping short, its about anticipating what is going to happen, controlling your speed and committing to a line. You'll develop good stopping power in your legs, but you'll also develop a quick reaction time and a keen sense of observation for traffic, peds, other cyclists.
When people get really good, its almost preternatural. I'm not there yet.