You should always keep your boarding stubs just in case.

We fly CW BA TATL (luckily out of the Caribbean so we get the elite miles) and though we've added our AA #s in the bookings, at check-in, etc. it's happened more than once that the registrations failed, with one segment or other ~ particularly some legs when we continue in CE.

We discard the stubs only after we've seen the miles on the AA statements.