User cannot edit one custom profile field other users can
Check the text format in use for the field that the user cannot edit.  Even if the user has permission to edit the field, the user will NOT be able to edit the field if the data is saved in a text format that the user does not have access to.

Make sure the user's field format is also filtered HTML.

If that doesn't work, use drush uli to log in as the user directly without changing his/her password and look at the problem with your own eyes-- sometimes users say something can't be done when they simply don't know how to do it.