[As moderator, I edited the thread title to make it more specific.]

This would be difficult to answer without knowing exactly what steps Delta took to book and confirm positive space for you on the AA flight. If your ticket was valid with positive space, you might have been involuntarily denied boarding if you were available for boarding by the cutoff time. Involuntary denied boarding requires a cash payment offer to you plus positive space on the next available flight. Since you were apparently not at the AA departure gate for the flight, you might not know if volunteers were first sought before involuntarily denying boarding.

When you went to AA to check-in, if AA had a "valid ticket" for you, did it offer you the next available AA flight to your originally ticketed destination?