With Amex it's fairly common for folks here to report that they got a card overnighted for no extra fee. 
It usually happens automatically if you get certain higher-end cards (PRG, Platinum, etc.) or if you speak with a CSR and specifically request it (also many times when CSR intervention is required on an application they will volunteer to overnight the card).
@Anonymous wrote:
I called them to check the status of my BCE and they asked if I wanted it rushed then overnighted it for free
I know that I said that having a card that said "Member since 87" was the least of my worries, and I wasn't going to worry about getting a new card with the correct MS date...but I was curious 
I went to look at the process of ordering a replacement card, and I was given the choice of regular mail or overnighting, both were free.
(And for the record, I didn't order a replacement, like I said, I was just satisfying my curiousity)