We have a very awkward and frustrating problem with our Magento store.
After several hours (sometimes days, sometimes a few hours) our site get broken (CSS and JS are not loaded anymore). We found out that it has something to do with the Magento base_url. When the problem occures the my CSS and JS are missing the prefix base_url. After I flush the cache the base urls are restored and the website is functioning normally again.
We use Lesti FPC.