At last, the mystery of Tom Bombadil's identity has been solved.
Tom Bombadil and the Witch-king of Angmar are the same person.
1. We never hear of Tom at all during the whole of the First Age. The Nine Rings aren't forged until the Second Age. QED.
2. You never see the two of them together.
3. In the first part of Fellowship of the Ring, the Nazgul are sent to the Shire to look for the wandering Baggins. Interestingly, Tom says to Frodo at the dinner-table: "...I was waiting for you. We heard news of you, and learned that you were wandering... But Tom had an errand there, that he dared not hinder" (Fellowship p.137 hardback, emphasis mine: note the fear Tom has of his master, Sauron!).
4. In Tom's questioning of the Hobbits, JRRT notes that "there was a glint
in his eyes when he heard of the Riders." (Fellowship p. 144) I think
he was concerned that his double-life might have been noticed.
Interestingly, Tom immediately changes the subject of conversation!
Furthermore, the One Ring had no effect on Tom - which seems consistent with Tolkien's observations about how the Nazgul would have handled the same priceless object (Letters of J.R.R. Tolkien, #246): "They were... in no way deceived as to the real lordship of the Ring."
5. It's also interesting to note that Tom could see Frodo clearly while Frodo was wearing the Ring (Fellowship p. 144 hardback) - just as the Witch-king could see Frodo clearly while he was wearing the Ring at Weathertop! (Fellowship p. 208 hardback)
6. Perhaps most damning, however, is the incident with the Barrow-wights
(Fellowship pp. 151-155), where Tom - with nothing more than a few
simple words (p. 154) - commands the Barrow-wight to leave. And it does,
without argument. Why would the Wight be so completely under Tom's
control? Because in his alternate guise as the Witch-king of Angmar, Tom
ordered the Wight to inhabit the barrow in the first place! Turning to
Return of the King, Appendix A, p. 321, "evil spirits out of
Angmar... entered into the deserted mounds and dwelt there." Obviously
the Witch-king was reponsible for sending the wights there; just as
obviously, the Witch-king (disguised as Tom) would be capable of ordering
them to leave!
(This is related to another passage, which has since been brought to my attention. On Fellowship page 158 hardback, Tom is guiding the Hobbits back towards the Road when he gazes towards the borders of Cardolan. "Tom said that it had once been the boundary of a kingdom, but a very long time ago. He seemed to remember something sad about it, and would not say much." Since Tom, as the Witch-king, was the one who destroyed the kingdom of Cardolan, it's little wonder that he wouldn't say much about his involvement. Perhaps his remembering "something sad" reveals some remorse at being the instrument of Cardolan's destruction...?)
...Yep: I think we have an airtight case here. :)
...It's worth noting that, after the Witch-king was dead, Gandalf said he was "going to have a long talk with Bombadil" (Return of the King, p. 275). Curiously, he never tells anyone about the meeting later... and he's right there at the Grey Havens at the end of the book, undelayed it seems by long conversation. I think we can therefore theorize that Gandalf made it to the Old Forest, but that Tom (once the so-called "Witch-king" had died) was nowhere to be found!
...Of course, all this brings up the curiosity of motive. What would make the Witch-King of Angmar sport such a double identity? I suppose that the Witch-king, once of proud Numenorean ancestry, felt trapped by the guise of evil which Sauron had tricked him into, and in the fullness of time forged this alternate identity for himself so that he could occasionally feel happy, helpful, noble, and more at one with himself and his lineage. The situation is perhaps analagous to a crossdresser who, feeling trapped in a man's body, would occasionally assume the identity of a woman. It therefore makes sense that the Witch-king's other identity would be so peculiarly enigmatic, and perhaps sheds light on JRRT's observation in Letters #144: "And even in a mythical Age there must be some enigmas, as there always are. Tom Bombadil is one (intentionally)."
...Who else would be aware of Tom's double-life, I wonder? Since Tom repeatedly claims to have been around "before the river and the trees", and indeed even claims to be older than the Ents (Fellowship p. 142), surely the eldest of the Elves would know he was lying. Elrond plays along with Tom in public, being kind enough not to reveal his secret, but also seems to know that Tom and the Witch-king are one and the same; hence his refusal to give the Ring to Tom for safekeeping (Fellowship p. 278-9): "Power to defy the Enemy is not in him."