Hi pinktank. Thanks for the post and sorry to hear you've been having problems; this definitely should not be happening, and the fact that it started occurring randomly is even more concerning.
Is it possible something on the site was updated that could cause this? Either The Events Calendar (ie 2.0.9 to 2.0.10), WordPress itself (to 3.5) or another plugin? If this is a byproduct of the 2.0.9 to 2.0.10 update - which I'm thinking it might be - then there is a possible explanation. We've had a few reports of this and it's due to the fact that we updated the jQuery used in 2.0.10 so that it was current and compliant with WP's standards.
As a result, certain themes/sites that aren't running the latest jQuery saw things break. Since The Events Calendar uses whatever version of jQuery ships with WP (as of 3.5, this is at least 1.7.2), any site running below 1.7.2 is going to encounter problems. It looks like in your case either your theme or another plugin is swapping the default jQuery for some older version, which is leading to these display problems. That possible?
Let me know if so. Please provide a link to the site, if possible, and I can take a look to figure out more.