Cadillac Repair: 1999 caddy deville starting problems., caddy, headlights

Question
I HAVE A 1999 CADDY DEVILLE W/ ONLY 58,000 MILES ON IT, I KEEP IT SERVICED REGULARLY. ON OCCASION THE CAR WILL NOT START . I TURN THE KEY OVER ALL THE LIGHTS ON THE INSIDE DASH GO OUT. IT CAN'T BE JUMPED. I HAVE TO LEAVE THE CAR ALONE FOR SEVERAL DAYS THEN IT STARTS BUT WHEN I LEAST EXPECT IT DOES IT AGAIN . THE DEALERSHIP CAN'T FIGURE IT OUT. HERE I AM 5,000.00 OUT OF POCKET DOLLARS AS TO NO WARRANTY. I'M GIVING THIS MY LAST EFFORT OF HOPE BEFORE I THROW THE TOWEL IN WITH MY CAR AND TRADE IT OFF. CAN YOU HELP ME?

Answer
When this fault occurs and you turn the ignition switch to the start position do you hear ANYTHING or do you hear the engine turning over but it just won't run ?.

Also try thid for me and let me know what happens:

1. Turn ON the headlights so that you can see that they are physically ON and then try and start the engine. What do the headlights do ?.