If you previously received a retention offer on an AMEX card, then when you try to close the card online, if it has been less than one year, customer service will warn you about it. If you still insist on closing the card, the bonus from that offer may be clawed back. Note that this one-year period is measured from the time the retention offer was given to you, and has nothing to do with the date when the annual fee was charged. Also, both downgrading and closing the card can trigger this claw back, so be careful.

So what should you do if you really want to close the card? My suggestion is to first wait until a full year has passed, then downgrade. The annual fee will be refunded on a prorated basis after the downgrade. After downgrading to a lower-tier card, you can then close the card, which maximizes the amount of annual fee you can get back.