Twitter does not archive all tweets. It does maintain an accurate count, but you cannot generally see all of them. So how to keep them?
Personal Tweet Archive:
Backupify -free for one personal account, up to 2GB. (Tweets are very small!)
Tweetbackup - free for personal use, requires you to follow them.
TwInbox - plugin for Outlook, puts your tweets into email, archives like email. (Ask your DSSL about this!)
Twistory - put tweets into a calendar.
Keeping Tweets from Others:
TwapperKeeper - saves tweets matching a hashtag
Read More:
10 Ways to Archive Your Tweets - ReadWriteWeb, August 2009