For Neo, moving faster than bullets was a matter of realising the Matrix's rules could be bent. For most of our video game heroes, bullet time is a popular mechanic that has appeared on titles like Max Payne, Red Dead Redemption, Vanquish and more.
It's not possible to slow down time and perform amazing feats -- not yet, anyway. Still, most of us have probably experienced that feeling of time slowing down. We know it as the result of a severe adrenaline rush. It's the closest we have to approaching bullet time in real life, and arguably what bullet time is based on. And there's a reason it happens.
Our adrenaline starts pumping when we're in danger, or when we're scared. What follows is that more information is committed to memory by the brain (the amygdala to be specific), making our recollections seem richer and denser than it actually was.
It's an illusion that tricks your brain into thinking that the more memory it has of an event, the longer it took to pan out. Time slows down for you, only it doesn't in reality.
When we get older, time seems to fly. This is based on a similar phenomenon to the one drawn above. We store more memories from childhood because every experience is fresh, and there's more information to soak up. Experiences we have when we're older are at risk of being lost in the fray, because we've racked up so many similar-looking ones already. This is why childhood seems to take forever and adulthood seems to zip by.
Giving us false memories, deceiving us into thinking time moves slower (or faster!) than it actually does: the brain is pretty crazy amazing, eh?