The only exception for this is the LHR-MAD flights that are operated by Iberia - you check-in with BA (well, all desks have both BA and IB logos), and baggage tags will say BA5xx.
On a SFO-JFK-ORY my tags got printed as AAxx and AAxxxx (when it was actually BA8006) and they never left T8. Just one fewer thing to go wrong, I guess.