I have gotten more than one sign-up bonus from Amex. But, it took a while. Many years ago I had an Amex Delta card which I canceled. Maybe 10 years ago. Got a sign-up bonus (15K miles?)
Then, maybe 3 years ago I got a pre-approved solicitation for a Delta Amex Card with an offer for 15K sign-up bonus and 30K miles if I charged $15K in six months. Sent it in, got the 15K, and the 30K bonus, too. Although I had to contact Amex Corp HQ to get the 30K. Not because I had a previous Delta card (never came up) but because they had somehow not entered my account properly for the bonuses.
I was happy.