Is it possible to brute force bitcoin address creation in order to steal money? Bitcoin users frequently generate new addresses for each transaction they make, which greatly increases the number of bitcoin addresses being used to receive money.

Would it be possible (and profitable) for someone to find collisions in the bitcoin address space in order to steal money?
It is possible to brute force some Bitcoin addresses, because some people generate their private keys in an insecure manner.  Any (non-zero) 32 bytes can be a private key.  So running sha256 over a passphrase gives an apparently random, but brute force-able private key.

Take sha256("sausage") for instance:

$ echo -n 'sausage' | sha256sum
30caae2fcb7c34ecadfddc45e0a27e9103bd7cfc87730d7818cc096b1266a683  -


Load up bitaddress and paste that private key into the 'wallet details' tab to get the corresponding Bitcoin address, then look it up on blockexplorer:

$ GET http://blockexplorer.com/q/getreceivedbyaddress/1TnnhMEgic5g4ttrCQyDopwqTs4hheuNZ; echo
0.01000000


and you'll see that the address held one bitcent for about 2 days in February 2012.

See also: "fuckyou", which held 2.5 bitcents for 12 festive days at the turn of last year.

So in practice it's possible to brute force bitcoin address creation, but only for poorly chosen passphrases.  These were probably just people playing around with the idea of "storing bitcoins in their head" which is why they are for such small amounts, and why they weren't left funded for long.

No address balances were harmed in the making of this answer.