The function file_unmanaged_copy in Drupal core calls drupal_set_message on certain conditions. Even with visible errors messages disabled, anonymous users see this. Indeed, the comment in the code even has a TODO to change it to an exception.

Obviously I don't wish to hack core, so is there a clean way to suppress this for anonymous and non-admin users?

Notably, the message is in a t() function, so I could use the stringoverrides module, but this would affect all users (including site administrator), so it's a bit problematic.