My car won't start sometimes...no sound..nothing. Other times it starts fine. What could be causing this?

I have a 1998 Dodge Neon that two weeks ago just wouldn't start. I had been doing some errands and hadn't had any problems with my car at all. When I went to leave from one of my stops, it wouldn't start. No sound or nothing, it just wouldn't start. I had it jumped but it did nothing...still wouldn't start. Tow truck came, the man opened hood and opened a box that had fuses in it pushed down on them a little, got in the car and it started right up. I drove it to my mechanic and showed him what the man did and he said sometimes a little corrosion can get under the fuses and cause it not to start and that it was fine now. And it was until two days ago. I went to start it Sunday evening and it wouldn't start again. This time it had a very low hum kind of sound only for like the first few seconds when I was turning the key, and then silent and it wouldn't start. I tried doing to the fuses like they did before but nothing happened. I tried it the next morning several times and nothing. Another tow man came out and it wouldn't start for him either, so he towed it to Goodyearmy mechanic. When they went out to bring it in the shop it started right up for them and has every since. They still have my car. They have checked battery, cables, cleaned terminals and tightened. They can't get it to Not start for them. I don't want to get stuck out anywhere at any given time. Do you have any idea what could be causing this?