What is stopping my theme from using a file that uses a suggested file name? I needed to modify a module's .tpl file and so copied it, renamed it to something suggested by Theme Debug's file suggestions.

Eg when I run drush theme_debug I see:

&lt;!-- FILE NAME SUGGESTIONS:
   * image-resp-item--f6.tpl.php
   x image-resp-item.tpl.php
--&gt;


So I rename my new file image-resp-item--f6.tpl.php and place it in my site's theme.

I clear cache and rebuild the theme registry but to no affect - the site still uses the original .tpl.

To experiment, I change my new .tpl's name to match the old one (remove --f6). This works - I see my new code appear on the site.

Why is my site (or theme?) not wanting to use the suggested file name?
Your can either use hook_preprocess_page or if you don't want to code use something like context module. You can add a new context to select theme template based on your requirement.