Unfortunately, such problems are rather common for WP. The general suspicion seems to be that it's more to do with the CloudFlare hosting service than with the site's own code - the protection in place against DDoS attacks seems very prone to false positives, and as you say, the robot-test often returns to the message editing page rather than posting (sometimes losing what has been typed - so it's always worth copying the message to the clipboard before clicking 'post').
Clearing browser cookies seems to help sometimes, but YMMV, and the problems usually return quite soon after. Quotes and other markup in messages seems to make the problem more likely, too, so they're best used in moderation, and removing them from a post can sometimes break the loop.
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When you are fighting an invisible monster, first throw a bucket of paint over it.