This has happened to me twice with AA and/or old US. I usually check a bag as my work often involves bringing along too many gadgets.
Generally they were very accommodating as they have a flat tire rule (if you are there within 3 hours, they'll rebook you on the next flight, even on a restricted ticket).
One time I was rebooked into SFO instead of SJC as it had a flight 30 minutes later, within the bag checkin time limit. Another time I missed the cutoff by 10-15 minutes. I basically used my (in the words of c-w-s
) mercurial charm to persuade them to accept the bag and expedite it to make the flight. I basically explained that I was late, explained what happened on the way, said I could really use a favour, and I had the US equivalent of a golden ticket to offer as thanks for their kindness. They did a quick override and called someone to deliver the bag to the plane. This was a couple of years ago when I had status on US, and I have no idea if it work today as a BA Gold, but AA generally treats me very well.
While your experience did vary, I find it far easy to resolve things at the airport if at all possible, rather than hoping to resolve things later, because it's unfortunately a lot easier for a phone agent that wasn't there to simply say no.