Wrong templates used for subpages whenever I edit /any/ page.

Whenever I make an edit to any Page on my site, trying to hit any subpage on my site pulls up index.php rather than the template which I have assigned to the Page. That is, I see my index page content when I navigate to /page/subpage. I need to reset permalinks to default and then back to any custom config to get subpage templates working again.

When I echo var_export($post) in index.php, sure enough, I see the info for the subpage to which I intended to navigate. So I know that the page is being called up but for some reason the correct template is not being applied until I reset permalinks.

I’ve had a look at wp_postmeta and there doesn’t seem to be anything awry with the template field there (it is still correct for those particular posts). My .htaccess file is read-only and contains the recommended WP settings only (paste of me catting it below):

<br />
$ cat .htaccess
# BEGIN WordPress
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
# END WordPress

I have been over “using permalinks” a few times and am at wits’ end. Any help would be appreciated!