WP won’t trigger 404 error

The 404.php template in my active theme is not called when the URI begins with “/index.php”.

http://example.org/index.php/not/an/entry
…results in an empty page with a useless “edit the entry” link back to itself.

http://example.org/index.php?error=404
…results in output of my home page content!

What does work is 404 catching when the URI does not start with “/index.php”:

http://example.org/not/a/page
…correctly gives me the 404.php page.

My custom permalink structure is this:
/index.php/%year%/%monthnum%/%day%/%postname%/

I use WP Super Cache (latest .71), but I get the same behavior with it inactive and its .htaccess additions commented out.