I believe you can only hold one. I had the basic BA amex and then applied for the BA PP card after triggering the 2-4-1; now when I log into the amex site the PP is the only one which shows, that I get statements for etc. It has the same membership year as the old card even though I'd held the old card for six months, the year did not reset. Essentially it's treated as an upgrade of the existing card rather than a separate card IME.

I'd spent the 2-4-1 before upgrading so I can't confirm on that.