How to edit theme files from WP admin

From the WP Panel I select Presentation, Theme Editor and bring up a theme file (page.php). At one point it says “This generates the subpage menu. If you don’t want to use it, delete this line.” I delete it and press “Update file”. WP says “File edited successfully.” but the line I deleted has automagically returned, i.e. the edit isn’t accepted. What am I doing wrong?
(I know I can fetch the file with ftp, edit it locally and copy it back, but why can’t I use this WP-internal feature?)