AA used to have quite a generous protection clause meaning that they would help if you had a delayed flight with them and missed a connection to another OW flight on a separate ticket. However, this was written out of their T&Cs several years ago. My experience has been, as mentioned above, that being non-expectant and unfailingly polite can help a great deal in this situation.

A little while ago, I was working in MIA and had flown AA F out and was flying AA F back. My wife decided to come out and spend 2 nights in Miami Beach (with me?) before I went home, but having flown AA economy on the way out re-confirmed her stance that she wasn't willing to fly economy back. Not being able to partake of what I thought was the very reasonable option of leaving her in Florida forever, and with the all of the flights from MIA to LHR completely booked up in J and F that day, I booked her an avios CW flight ATL-LHR, with a short hop MIA-ATL beforehand and a 6 hour layover in-between.
WHAT COULD POSSIBLY GO WRONG??
Anyway, a burst aircraft tyre, a missing engineer and 7hrs later she arrived at ATL. The BA flight was delayed an hour but had closed. While she was in the air flying to ATL, I was in the air using AA's shoddy Wifi and a contact that I dare not mention publicly dare I be hung drawn and quartered. When she eventually arrived at the gate, as God is my (her!) witness, the BA station manager at ATL re-opened the flight for her and let her board.

So these things can occasionally work out but my advice is, if you can avoid it, don't bother!