Apparently the U.S. LIbrary of Congress has them, the Tweets, all of them.. And here is their FAQ on the archive. The article and the FAQ are from April of 2010. I would assume that it is an ongoing project and that they are continuing to collect Tweets but I don’t know. They seem very interested in making the archive available to academic researchers:
What does the Library plan to do with the archive?
First and foremost, the Library is interested in preserving access to the archive for the long term. In addition to looking at preservation issues, the Library will be working with academic research communities to explore issues related to researcher access. The Twitter collection will serve as a helpful case study as we develop policies for research use of our digital archives. Tools and processes for researcher access will be developed from interaction with researchers as well as from the Library’s ongoing experience with serving collections and protecting privacy and rights.
But I’m having a little trouble finding out what the current status is, in terms of access. Actually, I’m not having trouble, I just haven’t looked really. It just isn’t obvious from these postings what the current status is. I will leave that up to you. :-)
Edit:
Here’s something a little more recent
This information on the LIbrary of Congress’ archive may actually not be all that helpful to you, if there is no system in place for yet for providing researches with access. It’s pretty daunting task, I imagine, managing all those Tweets.