Did (sort of) this once... had a ATL-DTW-AMS-ROM flight, with a short connection in DTW... plane was held in ATL for more than 50 minutes because they couldn't find "that piece that covers the inflatable slide". There was another flight to Rome leaving ATL that afternoon, and I asked if I could de-plane to take that one instead. I made it to the gate, where the pilot was asking where the mechanics were...

Long story short, he convinced me he'd make up the time lost in the air, after GA insisted that it was too much hassle to get my bag off the plane, so I went back to my seat.

Slept in DTW that night....