I'm a PM (not the DYKWIA type though ) and was delayed by mechanical on the flight from XNA to ATL last month. We arrived late but I thought I still had a chance at the connection, so I hustled to the gate and literally had the GA turn her back and shut the door on me after we made eye contact. I asked the other FA at the podium if he saw it, and he responded that he did but it's her flight and it's closed now. A group of three other pax showed up about 30 seconds after I did so I know I wasn't the only one who barely missed the connection.

I wasn't too bent out of shape since I had work to do and just headed to the SC, but what really got me was the flight from ATL to HPN. We boarded on time but the doors weren't shut even after our scheduled departure time. About five minutes later we were told that we were waiting for a some delayed connecting passengers, and sure enough about ten minutes later four or five passengers boarded the plane and we went on our way.

The inconsistency is what bugged me most, so I contacted Delta that night and explained what happened just as I've explained here. I got the form apology and a gift of 12,000 miles, so that took some of the sting out of it.

TL;DR: Sometimes, sometimes not.