Unfortunately it doesn't backup the complete set going back to the beginning if your account is too old or you have too many tweets (or at least that's how it worked last time I used it).

I ended up writing a custom script to handle the really old tweets.

For new tweets I use ITTT.com to send the tweets to WordPress. It works really well.

I put my tweets on a subdomain of my main site:
http://tweets.ryanhellyer.net/

It's really handy having them sitting there. I like that I can see the tags and the WordPress search facility is much better than anything within Twitter itself. It also means I have a backup, should Twitter go belly up, or lose data etc.

Another unexpected benefit, is that I actually get traffic from it. Google seems to preferentially send traffic my way instead of to Twitter itself sometimes.