This happened to me once at CLT. There were thunderstorms, and everything got backed up. As my LAX-CLT flight was approaching for landing, the aircraft right in front of us unexpectedly slowed down, causing our pilot to quickly pull back up, and we had to circle the airport. Due to the ATC backup, it cost us 45 minutes.

I had to run from the far end of the B concourse all the way to the RJ farm, and got to the gate just as my connecting flight to AVP (only slightly delayed) was shutting its doors. I tried to get the GA to keep the door open for me, but she was adamant. Just as I was about to head back to the US Club to get re-booked, she told me to hold on. She called to the pilot, who had agreed to let me on.....because there was a faulty wire which needed to be replaced, so the flight couldn't leave, anyway.

3 more people ended up boarding after I did, so I guess they left the gate open, knowing that a few more people might make it, after all.