I'd say it happens 30-40% of my flights into ORD that arrive at or before scheduled departure time (about 60% of total flights into ORD) and not infrequently at EWR. Can't say I've experienced it in IAD or IAH, and I don't get to the Western hubs often enough to have a valid sample.
A couple years ago I actually misconnected at ORD because our gate was occupied, the previous flight departed a few minutes late and the agent operating the jetway disappeared -- so we were on the ground ~30 early, but the door didn't open until +20 and I had to go from B22 to F16 or thereabouts. Got to the gate just as the door was slamming closed.