Given that the Tweet is already attached to an identity (individual or corporate account) and assuming one may claim even a reasonable semblance of creativity and originality on the content of said Tweet, it's technically Copywritable, but a mere "status" update, link, or factual listing is not.
- Think of all the copywritten corporate and government slogans (Just Do It, Army of One, What can brown do for you, Aim Higher, etc.) which could quite easily fit into a Tweet with room to spare.
- Think of all the haikus (see @MJ), other short poems, and unicode/ascii-art #twitterart (see @l_I__I_l) posts.
That being said, I can't think of any rational (e.g. economic) reason which would support registration of the Tweet as the "fixed medium".
Mostly, I can envision defensive sorts of legal actions such as blatant plagiarism or abuse/mis-use of a pre-existing logo, unattributed passage, or slogan.
As the multitudes of "sampling" lawsuit precedents in the music industry has shown, authorship rights of even nano-sized digital content can be successfully asserted.