We have a 2002 Chevy TrailBlazer. This vehicle has always had problems with the brake light bulbs not working. The problem we now have is that the 3rd brake light, or high mount brake light, works but the other two don't.
2002-2004 Chevy TrailBlazer tail and brake lamp problems resulted in a safety recall effective October of '04. The same recall also involved its GMC, Olds and Buick mid-size siblings. The issue was bulbs coming loose in their sockets, therefore becoming inoperable and a safety hazard. The repair procedure involved inspection and, if necessary, replacement of the left and/or right tail lamp circuit board assemblies.
But let's assume the recall was performed on your TrailBlazer and eliminate defective tail-assemblies from the picture -- for now. The third brake light (up top) functioning correctly eliminates the brake lamp switch at the brake pedal as a cause, because the same switch supplies all the brake lights with 12 volts of DC power. The third brake light coming on also confirms that fuse #12 (25 amp) located in the fuse block under the hood is good, and fuse #16 (10 amp) located in the fuse block under the left rear seat, which feeds power only to the third brake light, is also OK.
But in the same fuse block as #16 is fuse #34 (15 amp) which warrants a good look. This guy supplies voltage to the left and right side brake lamps. If it's burnt, replace it, but keep in mind that changing a burnt fuse usually doesn't fix the problem for good. The fuse will probably blow again shortly thereafter, and require a technician to trace and repair the short to ground which overloaded the circuit in the first place.
Also check to see if the directional and backup lights are working. That will confirm that their ground circuits, which are shared with the brake and tail lamps, are 100%. If you've found nothing to this point, remove the tail lamp assemblies and carefully inspect the bulbs and the circuit boards. Other than that, there's always the chance of an oddball break in a circuit due to a poor connection or a cut in the wiring harness.