My car does not like to fire up when the weather.

My car does not like to fire up when the weather is wet or damp and I can't find out why. I have changed the HT leads as well as the spark plugs and the car started fine when the weather was dry. However, whenever the rain, fog or mist come along the car refuses to start. I have rang my breakdown company twice about the problem and both times the mechanics have come out but haven't actually been able to pin point the problem other than tell me to change the HT leads and the spark plugs in which I did and it still doesn't want to fire up. Can you tell me what else it might be? Or do I need to take it to a mechanic where I live and see if they can find the problem?

Thanks