The Amex Once-Per-Lifetime Rule and Pop-Up Jail
How once per lifetime works
American Express ties each welcome bonus to a specific card product. Once you have earned the bonus on, say, the Amex Gold, you generally cannot earn it again no matter how long you wait or how many times you cancel and reapply. Different cards are separate, so earning the Gold bonus does not stop you from earning the Platinum bonus. One important catch sits on top of this, though: a separate Amex family rule blocks the bonus on a lower card in a family if you already hold a more premium one, so while Gold does not block Platinum, holding the Platinum does block the Gold bonus.
Pop-up jail
When you start an Amex application, the site often shows a message saying you are not eligible for the welcome offer on that card. People call this pop-up jail. It is Amex telling you up front that you have had this bonus before or are otherwise ineligible. If you see it, do not submit, because you would get the card with no bonus.
How to play it
Treat each Amex bonus as a one-shot. Apply when the offer is strong and you actually want the card, since the bonus you take is the only one you will ever get on that product. Check for an elevated offer first, and back out if the pop-up appears.