The profile field is a very simple long text, text area, filtered html. When I try to edit it in my own regular user (no admin roles) page, with the standard permissions everyone has, I can do it no problem. A specific user has reported that he cannot do it at all. A message is displayed that his permissions are insufficient instead of the editable text area. A third user was kind enough to click his own edit button and has no problem just like me.
I double checked permissions (field permissions module is in use) and they seem absolutely ok. I tried to peek inside the database but don't see anything suspicious (I may not be looking in the right tables).