Welllll. I finally got the W124 I just bought ("Elaine"

) up and running, so tired, worked on her this entire week. She drives, and feels... Freaking... AMAZING!!! It's literally the W124 of my dreams, EXACTLY how I wanted my old (in my sig) W124 to be JUST like.
I've encountered a small problem though and had her towed back to my garage..

. While I was driving back from work, the RPM's slightly hesitated and she turned off. I turned her back on and the same thing happened after a while, until finally she just wouldn't turn on anymore. I would of sworn it was the fuel pump because the symptoms were exact of a bad fuel pump.
I just jacked her up and checked out the fuel pumps (while they're connected to fuel lines) with a battery. They BOTH work, and when they turn on there's pressure on the fuel line! I immediately thought the relay might be bad or something electrical. When I turn the key to on though, I can hear them humming, and she turns on for a few seconds then turns off. I plugged the fuel pump (just one) to the battery and turned the car on. And she stays on perfectly fine, as if nothing ever happened. What the heck? :S
Is it possible for a fuel pump relay to kind of "go bad" while the car is on and then work the next crank? What do you guys think?