Diagnosis
Yeah, this one's common. The hole is small and grease cools and solidifies quickly, especially if it has to travel across the cool bottom plate to get to it.
How to Fix It
The best fix is to create a better path. Get a small, disposable aluminum drip pan (like a loaf pan or a small steam table pan). Place it directly under the drain hole on the bottom plate. The grease drips into the pan instead of running across the metal. When you're done, just toss the pan or empty it.
If That Didn't Work
Some people drill out the drain hole to a slightly larger diameter (like 1/2 inch) and attach a short section of metal pipe or a nipple as a spout to direct grease straight into a bucket. This is a more permanent mod.
Heads Up
Don't let grease pool in the bottom of the cook chamber—it's a fire hazard if it gets too hot and ignites. The drip pan method is the safest and easiest. Just make sure the pan isn't directly under a huge heat source from the firebox.