There are multiple datapoints coming out of American Express shutting down credit card accounts, it seems those affected are having all of their cards closed (sources: /r/churning, reader e-mails, @milesperday and MtM). Those with accounts shut down report receiving an automated e-mail message informing them of the shut down. The common thread seems to be cardholders that have used their cards to make a large volume of payments using PayPal Key, the shut down risk might also have been increased if this was paired with the 3x referral bonus.
In the past American Express has shut down accounts for ‘abusing’ no lifetime language links and other reasons.