Theory: Tesla's machine never worked. Nolan tricked us
This theory is based on the idea that Nolan's writing is far too clever to simply end the movie by actually throwing in real magic (a cloning device) and spelling it all out for the audience at the end.
In fact, it's mostly a lie.
The cloning machine never worked. We're simply fooled by the way the story is told.
The entire themeof the movie is multiple layers of tricks, and as such Nolan is ending the movie by pulling a Prestige of his own on the audience.
I think the theory is best explained by going through the events.
Timeline of what I think actually happened:
1. Movie starts with Angier hating Borden and being obsessed with the Transporting Man trick, extorting Borden for the secret.
2. Borden decideds to pull a major hoax on Angier by luring him to Tesla. They trick Angier is tricked into thinking the Cloning machine actually works, by planting a bunch of "cloned" hats and cats for him to find.
3. Angier tries to clone himself but never gets it to work. Figures out he's been duped.
He now gets completely obsessed with revenge and decides to start investing his entire life as part of the trick (same way Borden does)
Decides pull the Ultimate trick on Borden: To actually make him believe magic is real, by tricking Borden into thinking the cloning machine actually worked! [evil]
4. Angier hires a lookalike and starts performing his own version of the Transporting Man.
5. He knows Borden will try to unveil his solution, so he sets Borden up - by killing his lookalike in a water tank in order to frame Borden for the murder.
6. Angier becomes Lord C and starts writing a journal, intended for Borden's eyes only. He writes down the lie that he actually did manage to clone himself.
THE KEY HERE: This movie uses some subjective perspectives. What you see isn't always what you get.
The largest part of Angier's story in the movie is actually told through Borden reading Angiers journals.
So scenes where Angier clones himself never happened - they just play out as Angier wrote them in his journal and the audience sees the same thing Borden envisions when reading the journals.
7. Borden is now jailed but Angier figures he has an accomplice that'll get revenge some day. To screw with Borden's mind, he places a bunch of well crafted wax dolls in water tanks, to appear as his "dead clones". [evil]
8. Borden's twin finally Angier. In his dying moments he talks about how he dedicated his entire life to this final trick.
And the final trick was this elaborate plan with the clones, which convince Borden that it wasn't a trick and that real magic in fact exists!
At the same time, Nolan also pulls this final trick on the audience.
THAT is the Prestige.