CSS aggregration breaks site - how to troubleshoot?
Some initial things to check:


Clear the cache (if you're running advagg, clear its cache). You can end up with stale file references sometimes.
Find out what's happening to the missing CSS:


When the styles go missing, does the page source contain a &lt;link&gt; to an aggregated CSS file? If not, check for modules/themes implementing hook_html_head_alter(), hook_css_alter(), or hook_preprocess_html(), and see if they're doing anything strange.
If the source does contain the &lt;link&gt;, what status code is the browser returning for the request? 404? You might have a stale cache issue. Clear all caches including those on any reverse proxy you're running. It might easily be something else, would require more intimate debugging. 500? Check the server logs, that should give you clues as to where things are failing.
If it's 200, does the requested file actually contain the styles it's supposed to? If not, you might be looking at a cache issue again, or potentially an issue with build tools (Compass/Gulp/Grunt/etc).

Switch the theme to a core one. Does it have the same problem? If so, it's probably a module or server setting at fault. If not, it's likely to be a problem with the theme.
Make sure permissions for your files folder are at the right level - the web server must be able to write to that folder to store the aggregated files. This is less likely to be an issue if the files are being created successfully, but it's worth double-checking you don't have an intermittent file system issue (if it's over NFS for example)


Getting the answers to those questions will help to narrow down the source of the problem, but unfortunately it's difficult to give specific advice as to where you should look in the code, as there are a lot of points problems could occur, and a lot of different modules you could have installed, at varying versions.