In general, a paintball gun can fire as fast as you can pull the trigger. Some paintball guns have long, hard trigger pulls that the effective rate of fire is about 5-6 shots per second. Other paintball guns are fired with circuit boards and can fire up to 30 or more times per second. This, though, is only how fast the guns can cycle and not how fast they can actually shoot paintballs.
The problem comes when you try to load paintballs into the gun. A gravity-fed hopper is limited to about 8 balls a second while some motorized hoppers can feed over 20 balls a second. Many high-end and lower-end guns are now capable of shooting as fast as any hopper can feed the paintballs.
Additionally, many people can't move their fingers fast enough to fire more than about 10-12 BPS. Many players claim that they can fire over 20 BPS in semi-auto, but very few can actually back that up on a timed chronograph. Extreme rates of fire are almost always dependent on ramping or fully-automatic firing.

