The most common reason that the rewritten resources 404 is because of mod_rewrite RewriteCond rules. For example:

RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^ /404 [L,R=301]


This rule causes 404 for all requests which don't exist on the filesystem, including mod_pagespeed rewritten resources.

In order to fix this you must add an exception for mod_pagespeed URLs:

RewriteCond %{REQUEST_URI} !pagespeed

This will allow rewritten resources, the admin console and static resources necessary for some filters.