I was getting a
Page Not Found error when I tried to edit my profile.
Incidentally I logged in to my Area 51 account, tried it again, and it worked, so it likely only happens when a user is logged out of the main area, but logged in on the Meta site.
I assume this is not standard behaviour, but on other pages on the Stack Exchange network this state of being semi-logged-in is impossible, as far as I know, so this could be an oversight more than a bug.
Side note: there is also no tag for user-profile or something similar, it seems, and I can't create one.