Jimbad

This intermittent problem has now happened for the fourth time on my 2003 with 4,700 miles on it. I do nothing but open the door and the chime goes off! When I turn the key nothing happens; complete silence. The battery (brand new Optima) is in absolutely top-notch condition. The first couple of times I got the car started by tapping on the ignition relay. Coincidence? This last time I had it towed to the dealer in its non-starting condition, per the tech's suggestion. This dealership's (Criswell, in Maryland) Corvette techs seem to be very good but they've had the car a couple of days now and I am getting nervous.
I am hoping that some member(s), with their vast experience, can help solve this nagging problem.
Is the chime going off any clue?