Although it is has not been brought up much, if at all, in studies of Tolkien, the Nazgûl are one the major examples of an internal inconsistency in Tolkien's works. While part of the attraction of Tolkien's work is the fact that there are many aspects of it that aren't neccesarily part of his main narrative arc, there are still some ground rules, mostly of a theological nature, that Tolkien seemed to hold fast to.
Stated explicitly in The Silmarillion is that Humans have the Gift of Erú, meaning that they die.
Although it is something that humans fear, it is something that is actually for their benefit. This is a divine right, and no power on Earth, even Manwë himself, has a right to revoke it.
However, Sauron is able to devise a device, the Nine Rings (as well as, for that matter, the One Ring), that gives people seeming immortality. In other words, Sauron, a corrupted Maia, has managed to defy the Will of God. From a theological point, this would seem to be unacceptable to Tolkien.
Of course, the rings don't actually give more life, they merely stretch it out. However, even though they don't give life, they do manage to take away death. I can only think of two ways that this would be permissible in Tolkien's worldview. First, it could be possible that the rings don't grant immortality, but merely stretch a person's lifespan by a hundred fold, but that even the Ringwraiths will one day die. Another possibility is that the wraiths contain the intellects, wills and memories of the men who once wore the rings, but that their souls have already fled to the Halls of Mandos. A story external explanation is that Tolkien just did not think this aspect through clearly.