In case you didn't see Christian Bale and Wolverine... um... Hugh Jackman in The Prestige yet, stop reading. I am going to give away a major plotline here. Don't blame me if you keep reading... you've been warned.
As far as I can tell there are only two scenarios that make sense with this movie.... two explanations that neither of which are entirely satisfying.
1) Christian Bale has a natural twin that he has grown up with all his life. He makes "Telsa" the key to his book because Telsa is attempting a cloning machine that doesn't work, sending Hugh Jackman on a wild goose chace. (Because, if he was aware of it working, he would have essentially given his one advantage, being a twin, over to his arch rival.) By sheer dumb luck, the machine actually turns out to work. Now, actually, I believe Tesla was more likely trying to invent a transporting machine... and that happens to be the wackiest glitch in the world... that it doubles you. But, either way, Bale could have never thought it to work.
2) Telsa actually did make the machine work for Bale and he used it once, to clone himself a twin. The second Bale doesn't seem to exist very early on in the movie, like when he's a stagehand. Where was he all those years if they weren't doing the magic act routine? Plus, it's just too random that the key to the diary is Tesla, a man that, in reality, has nothing to do with his trick, b/c Bale already has a twin and doesn't need a transporting or cloning machine... of which Telsa actually winds up, ironically and accidently, creating a cloning machine that works. Of course, this doesn't make sense either, b/c Telsa doesn't even know his machine works until Hugh Jackman comes to visit him. If that was the case, though... why would Bale have anything to do with Tesla?