Is the bird killing in The Prestige based on historical reality?

It is based on a real trick, but the sources I provided do not conclusively state if any birds in these acts were ever killed. Only rumors and at least one inquiry, resulting in no substantial proof.

It appears the general nature of the trick in The Prestige is a variation of the real life, Vanishing Bird Cage:

The vanishing bird cage, also known as the flying birdcage, is a classic parlour magic effect that was invented by French magician Buatier De Kolta. The trick has also been used by magicians Carl Hertz, Harry Beardmore, Harry Blackstone (Sr. and Jr.) John Mulholland, John Angel, Sabrina Vera, and Tommy Wonder.

The magician displays a bird cage, holding it between both of his hands. The cage is rectangular, about six inches tall by six inches wide by eight inches long, and made of wire on all six sides. Often there is a bird, though in modern performances of the act it is usually fake, inside the cage. The magician will offer the cage for inspection by an audience member, but he will never actually release his grip of it. Then, without covering the cage, the magician makes a sudden motion and the cage (and anything inside) vanishes from sight. A variation of the trick was featured in the 2006 film The Prestige.

More Information:

The Vanishing Birdcage was a sensation and was also quickly ripped off. In the Summer of 1875 Harry Kellar is said to have purchased a cage from DeKolta's cousin for $750. Of course this was unauthorized because Buatier never sold cages to anyone. This cage was probably the very first one outside of DeKolta's act, but it wouldn't be the last. In fact, Harry Kellar can probably be credited for the deluge of Vanishing Birdcages in America because he sold the secret to a magic dealer in exchange for props. In Europe a letter from Robert Heller to Charles DeVere the french magic dealer shows that the cages were already for sale in December 1875.

Harry Kellar stirred up a bit of controversy while in Australia over his presentation of "The Flying Cage" as he called it. Harry Kellar's routine was simple and direct, he counted to three and the cage with a live canary inside would vanish! A rumor circulated that Kellar was killing a canary every time he presented the effect. An inquiry took place and Kellar proved that was not the case. He showed that he had one bird and one bird only that he had been using for a long while. But this same controversy would come to haunt other magicians across the globe. In fact, this controversy was used as a minor plot point in the movie "The Prestige" in which they give a rather fictitious explanation on how the cage works.

I did a little research on this question and it appears that the trick MIGHT kill the canary, or might not, but inevitably did result in a lot of bird deaths. Here is the notice:

The vanishing bird trick (which, I regret to say, kills a great number of canaries) is performed by means of accessories in which the costume plays a part. The bird cage itself is made to collapse into a cigar shaped tube. The centre of the cigar shape is naturally fatter than the ends, and, if the bird is fortunate, it is secreted there and may come out of the ordeal alive. Should, however, the bird get at either end it is crushed to death, or should the legs get through the bars they are almost certain to be broken. A conjurer once told me he had done the trick 200 times with one bird, but I should very much doubt it.

What happens is this. The bird cage is rapidly hooked to the end of a piece of whipcord, which passes up the sleeve and across the body of the performer, and then down the other sleeve, where it is tied to the wrist. It will thus be seen that the cord is considerably shorter than will be required to stretch from each end of the arms if they are extended. The cage, being in the performer's hand, is rapidly compressed and made to assume its cigar shaped form, and is dragged up the arm at lightning speed by the simple process of extending the two arms.

This very simple trick has puzzled millions of people. It is performed so rapidly that even if you closely watch the conjurer I doubt if you will know what he is doing. Afterward the conjurer usually pretends to find the bird among the audience. As a matter of fact he takes another bird out of his pocket, and by one of the simple dodges of conjuring appears to take it from some one's neck or head, or some other convenient place.

— Current Literature, Volume XIII, May–August, 1893, page 230