I had 5 personal (non-charge) Amex credit cards and 1 PRG Amex at the beginning of the year. When I later on applied for the Blue Cash Preferred, Amex asked me to close two non-Charge cards so they could approve the BCP card, now limiting me to 4 personal non-charge Amex cards.

I have two other third-party Amex cards (Citi and Penfed), which of course don't count towards the Amex limit. I'm actually considering opening other third-party Amex cards just to take advantage of the Amex sync cashback offers.