[Telepathic Intruder]

A space elevator won't have to reach the ground either. The entire mass, cable + asteroid, would be centered in geosynchronous orbit. In the case of this article, it's still geosynchronous but not stationary because the orbit is inclined to the equator.

Carbon nanotubes are supposed to have enough strength, so they say. But what if it does snap because of a terrorist attack or a meteor impact? The cable would wrap around the earth as it falls, delivering the impact energy of a nuclear explosion where it hits.

[Boogieman]

I would be more worried about the giant asteroid impacting the earth when its orbit inevitably decays.

Maybe the dinosaurs tried out this plan before we did :)

[Ozark Tom]

Change the load mass (building), or distribution of the mass, within the space elevator system, and the orbit will change.

With enough velocity, the tether would vaporize while transiting through the atmosphere.