Car won't start in cold + damp weather! Help!

I have a 97 Jetta, and when it it is cold and damp out my car will not start. At times, if I persistently try to get it started, it will finally turn over after 5-10 minutes of periodically turning the ignition and pressing the gas pedal…though sometimes it won’t start at all. Once it is running, the problem won’t return during the same day. I have had a recent oil change, new battery, and live in SF California where the temperatures never get below freezing. I brought it to the mechanics and they didn’t find anything on their diagnostic tests. I don’t know much about auto engines, but my instinct is that the fuel isn’t getting through/ignited? Help!?

Please be more specific about how it doesn’t start. Does it crank but not start running on its own? Does it not crank at all? I suspect the former, since the problem happens when it is cold and damp. Usually this is caused by a failure of the high voltage part of the ignition system, like the coil(s), plug wires, and/or spark plugs. Your mechanic knows this too. Since your car is 12 years old, I’d start with new spark plugs and wires. And distributor cap if your car has one.