In the opinion of intellectual copyright attorney Brock Shinen, the short answer is "no".
I admit, I think a protectable Tweet exists in theory. I have read hundreds if not thousands of Tweets and have yet to read one I believe would be protectable, but the possibility exists. The question is not: Are Tweets Copyrightable. The question is: Is This Tweet Copyrightable. The copyrightability of Tweets is not dependent on the fact that they are Tweets. Rather, it's dependent on the analysis of the Tweet in question. The all-encompassing response that all Tweets are either protected or not protected is misguided. The real response is that it depends. However, when you analyze most Tweets, they would never individually pass copyright muster.
Does this mean that nearly all of Twitter's content is in the public domain? Or can you copyright a collection of tweets...the entire output of one person, for instance? Let's say I want to publish Tweatise: The Wit and Wisdom of Merlin Mann, an unabridged book of Merlin's Twitter stream...can I do that?
Update: Another opinion: tweetCC.
tweetCC makes it easy for you to offer your tweets under a Creative Commons Public Domain Dedication or other Creative Commons licenses.
(thx, dan)
Update: Yet another perspective.