2003 Buick LeSabre Electrical Problem

I have a 2003 Buick Lesabre. Purchased with 89K miles and problems started at 120K miles. It ran great for 18 months. Still runs great - when it starts. I will drive it wherever, run errands, fuel, etc - and it wont restart. It has done this 4 times since December. I turn the key and nothing. No sound, no click - radio, lights, AC, electric windows, etc - all work. It has a new battery and been towed twice to two separate dealers. It starts fine when they have it. They kept for three weeks and start everyday, no problem. Two times it has restarted after about 10 minutes, I have looked for fuses under back seat and ignition relays, etc. Don't feel like that was the problem due to primary ignition being under the hood. Any thoughts? Thanks.

What is security light status when in "no start" condition-- is yours still a chip in the key? I believe so-- so what I do is wait 4 minutes until light goes out (using this 4 minutes to "polish" the resistor on my pants leg) and then try again- with success...

How about trying "the other key" does that help when in "no start" status?

When it doesn't start, besides checking for that security light, like brcidd said. ... when trying to crank, see if the lights dim.

If the starter is in fact trying to start, dome lights, dash lights, headlights (with a buddy looking ) will all dim. If they do not, you need to check your main battery wiring. Both ends of both cables. Sometimes the contacts on the ends are corroded inhibiting tight connections... lights, radio, accessories aren't a big enough load.
Please update us when you've done more checks or you figure it out. Looking forward to hearing from ya!