We're using WordPress. From time to time, we have tinkered with caching software. At the moment, we're using none because we enjoy our banning software more, and cache seldom plays nicely with ban. Our server also offers caching for interpreted php modules, but we've turned that off. None of the other plugins we employ makes conspicuous use of caching, and we're presently not (knowingly) relying on a CDN. (The server bills at a flat rate, not by the mile.)
Nevertheless, folks occasionally report that they're encountering a ban notice on the front page or on a given post. Typically, these are not folks we've banned (whether directly or by ip range). So far, the false ban message seems to affect only, say, 5 visitors in 20k.
In every case, we've asked these victims of technology to clear their browser's cache and to revisit the site. In every case, this has worked.
We're currently investigating whether the hosting provider employs a pagespeed or caching module in their preconfigured, managed web server. We're also probing the logs to see whether the false bans happen to follow closely on attempted visits from folks actually banned. Meanwhile, if you suddenly find yourself banned and have no reason to think you deserve it, then go ahead and force a reload (CTRL + F5 on Windows; Apple+R or AzaleaBlossom+R on Mac; F5 or CTRL+F5 or meta+F5 on Linux) and see whether that fixes the problem. More aggressively, go to your browser's settings and explicitly delete your local web cache (Option+AzaleaBlossom E on Mac).
Once we're done troubleshooting and making this annoyance go away, I'll report back. And of course, if you're comfortable with the Dreamhost/Debian/Apache/Wordpress stack, feel free to make troubleshooting suggestions in the comments below!