It used to be possible to refer yourself to an American Express card and get a bonus as both the new member and as the referrer, but then AmEx began clawing back points earned from such referrals. Additionally, people who engaged is such activity found themselves unable to use referral links at all, even to refer friends. On new cards, however, people found that their referral links were intact and able to be used – it was only on existing cards that were there at the time of the self-refer ‘violation’ that lost their ability to refer friends.
However, multiple reports on Reddit (1, 2) indicate that yesterday even new cards stopped being able to refer, and all existing referral links from those cards no longer work. There’s one report of someone losing their ability to refer, even though they did not engage in self-referral, so there might be something else at play here as well.