An easy to configure Twitter Plugin with Pretty URLs.
There are a couple of reasons that this may happen. One is that Twitter's API rate limit has been exhausted. The rate limit is refreshed hourly. If you are on a shared server this can be caused by someone else's site.
Another possibility is that your Twitter account is protected. Private accounts cannot be accessed via the public API.
A future version of the plugin will include an option to register a private API key. This will prevent both of these issues.