Many Amex CCs has now in t&c for eligibility of signing bonus:
"If we identify you as currently having an American Express® Card account, you may not be eligible for this welcome bonus offer."

Actually as a current Amex card holder, I receive this message when clicking on any cc offer:
"You have been identified as a current American Express Card Member. You may have seen an offer advertised for this product previously, but the offer is not available to current American Express Card Members."

Now I really feel better doing business with Chase.