Are you referring to the private key which is generated the first time a person runs Bitcoin? If someone were to successfully duplicate someone else's key then after they downloaded all the blocks, they would have the same balance as the person with the original key. That's what you're saying, right?
Yes, that is correct. They would share the wallet, and it becomes a race on spending the money first.
I was referring to the custom Bitcoin addresses which you can label with the name of the person who is going to send you bitcoins so that you know from whom the payment came. I think these addresses are generated from the private key mentioned previously. I'm wondering about collisions because although they are very unique, they are easily generated over and over again by all by all bitcoin clients.
If I read the source code correctly, keys are always made in pairs. That means, every address has an associated private key. When you click "New Address", you call GenerateKey in main.cpp, which generates a new key pair. So the duplicate address is ultimately a duplicate public key. Which is
very unlikely.
While keys still are "easily generated", you should have to generate a whole lot of keys before a collision. While I am not certain, it seems that the keys generated have a space of 256 bits, which is a lot more than the 122 bits put in perspective in the wikipedia article on uuids. Remember, 123 bits have half the probability of collision as 122 bit, 124 half of that again etc.
