It appears that if you are using Style 10 within the Twitter module in a layout, if the twitter feed does not contain any tweets the handle (which normally appears on the bottom) does not display.
Template 10 if there are no tweets returned looks like this:
I’m not sure if we can get the twitter handle for the account or in other templates the account logo etc to show if there are no tweets to return, but perhaps that’s something we could look into in the future.