Hi Evan.
In my previous work life, I made use of the fact that most CPUs never go totally to 0.0 volts, they usually will be above zero and well under 0.5 volts. When I was trouble shooting a CPU or logic problem, if I measured true zero volts, I looked for a short. If I measured anything between zero and 0.5 volts, I looked for a logic problem. And if I still couldn't find the problem, I'd finally check the voltage at both ends of the PCB trace just in case it was open. It didn't happen often, but once in a while the pin would look soldered but would be floating above the matching pad insulated by a bit of flux. I can't begin to guess how many times there was a micro-thin short between pins or traces that I couldn't see even under the microscope, but washing away the flux residue and cutting between the traces would solve the problem. But I'd also say you are right to be careful before getting too frisky with the Exacto knife if you don't have good magnification to work with. :-) When I was doing repairs, I could always tell when the previous tech was using a 3x bench light instead of his 10x microscope while doing his soldering. The better the seeing, the better the work on small SMD parts. Good luck with the repairs.
Doug.