You should check out some videos of CIWS (Close In Weapon Systems) in action. They're systems designed to shoot down projectiles like missiles and mortar rounds (as well as targeting small vehicles and planes). The sheer number of rounds they spray to take out a target that is moving on a single ballistic trajectory is crazy.
The closest thing I know of to what you're talking about would be hard-kill APS (Active Protection Systems). These are systems designed to protect vehicles like tanks from incoming rounds and missiles. Using radar and optical sensors, they can detect a round and predict whether or not it's going to hit the vehicle and respond in nanoseconds, firing an explosive back at a target traveling 1-2km per second. However, this isn't like shooting a bullet out of the air with another bullet. It's more like chucking a grenade at a missile to either deflect it or destroy enough of it that the pieces (still going 1-2km/s) don't damage the vehicle.
But both of these systems are designed mainly for destroying targets on a ballistic trajectory. When you throw drones into the mix, now you have a target that can react to your reaction. With slower moving drones like the helicopter ones, that's easy enough. But what about a drone that's moving at mach 2 and capable of sustaining 20g's, like a missile. Now you're talking about basically firing missiles at missiles, which has proven to be very difficult before a missile has spent its fuel and is coasting towards its target on its final ballistic trajectory.