On a personal/free user account, I can create a single access token just fine at this very moment. Trying to create another throws 403 Forbidden in the browser’s network console, along with the dialog shown in the first post.
Disabling the existing token allows me to create a new one, but unlike what I described in Cannot create New Access Token, button greyed out? two months ago, I cannot enable more than one at the same time right now. Trying to enable the inactive one gets me a 403 Forbidden as well, along with “This token could not be updated at this time. Please try again later.”
So, @sforbes I assume you already have one or more existing tokens too? (Which is not to say that this is not a bug.)