At least for domestic, in terms of a checked bag, the delayed departure time "resets the clock" as to the cutoff time.

Echoing ethernal's post, BDL is my home airport, so when an inbound is delayed from DTW/ATL/MSP, there's a pretty minimal chance that anything is going to change, so I don't bother rushing to the airport. All of the BDL ticket agents have said that when that happens, I'm in the clear to drop the bag off later, as long as it's by the cutoff before the delayed departure time.