It seems that they do. At least the Witch-king, the most powerful of them does.
Quote:
[The Witch-king] bent over [Éowyn] like a cloud, and his eyes glittered; he raised his mace to kill.
But suddenly he too stumbled forward with a cry of bitter pain, and his stroke went wide, driving into the ground. Merry's sword had stabbed him from behind....
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The Battle of the Pelannor Fields (emphasis added)
Presumably that idea was carried over to the movies. At least I don't recall anything which would conflict with it.