In this Nature.com article, the author cites a study by Nicola Pugno of the Polytechnic of Turin, Italy, which indicates that carbon nanotubes are not strong enough to make a Space Elevator because of their “inevitable defects”. And, he says, even if the defect problem could be overcome, “damage from micrometeorites and even erosion by oxygen atoms would render them weak. So can a space elevator be made? “With the technology available today? Never,” he says.”
This article has, obviously, generated quite a stir. Some of the debate can be seen over at the Yahoo Space Elevator forum.