I'm having a problem with full page caching whereby when a customer enters a product quantity which is more than the available stock, it sometimes alerts the user on the product page, but other times it doesn't alert the user until they check their basket.
In this case your global messages block has not been properly cache busted. You have likely relocated it or you are calling the echo for the block not by alias, but by calling the core/messages singleton directly.
To rectify, follow the standard debug procedure: change your store's theme to the base/default theme and follow your test cases again. If the problem persists, contact Enterprise support as there were some known issues with certain blocks and page cache in 1.11EE.
If you're on 1.12 or above the block is properly holepunched.