I don’t know if you’re on a shared server or if there’s another plug-in that’s hitting the server very hard, but you appear to be ‘rate-limited’. (Your server is hitting the Twitter servers too frequently.
This can lead to your server getting blocked from reading Twitter for quite a long period of time.
There’s a comment in your HTML that reads:
<!-- Rate limited -- Next attempt to get data will be in 58 minutes -->
More about rate limiting at https://dev.twitter.com/docs/rate-limiting.
The only thing I can suggest is loading your site at a time when the server might be hitting the Twitter server less hard (early morning?).
If you look at the HTML for the rotating tweets, it will tell you how long to wait before your next chance to get data.
Even when it wasn’t being rate limited (as in tweets were showing, but not new ones) it wasn’t quite refreshing, but now it is. So whatever magic you have worked has done the trick 🙂
