Like Tunafish eluded to, running a motor only requires three things: Compression, fuel, and spark (although he said air).
Compression is generally never suspected when an engine will run before and after an intermittent problem (in fact compression is generally the least likely suspect of the three).
So from a mechanic's perspective, you're left with two suspects: Fuel and spark. Years ago when I was a backyard mechanic, we would spray Starting Fluid into the carb to eliminate spark as a suspect. If it starts after spraying, then you know you have a fuel problem. It will only run for a few seconds...but tells you what you need to know. Not sure how people feel about using Starting Fluid on this forum...
EDIT:
Here's a video of what I'm talking about. But always keep in mind that starting fluid is extremely flammable. You should always assume the worst when using it; ie that there will be a backfire, etc. Keep a damp towel on hand to smother a fire and, as a last resort, a fire extinguisher.
