Drupal 7 website layout/styling breaks all a sudden (random), how to debug? there are no logs
Above all, I assume that you have Google Analytics Installed. Go to the Behaviours menu and from there look at Site Speed and Page Timings to see how much times it takes to load a page. Use the Speed Suggestion option to allow Google suggest you a number of fixes/tricks that can potentially improve the performance of your web page. This is very handy. 

This is clearly a problem of CSS. Open your site in chrome, enable the development tool F12 and then keep refreshing and monitor Network and Console windows. I observed that every reload of the page causes a request to a CSS to break (hence the resource is never downloaded - each time a different CSS from a different module). 

It would be a great idea if you could give this module a go https://www.drupal.org/project/advagg, I highly recommend that you aggregate all your site's CSS into few or single CSS file. 

It is always better to download one slighty larger file than make over a hundred calls to a web server requesting for tiny tiny CSS files, because there is high chance that your request fail or die (to to waiting). 

I will update my answer with more findings but, give that module and a go and let us know your findings.