Assume that person wanting to be anonymous has reason to do so. Powerful enemies will see what is bought with bitcoins, perhaps web blog on server and domain name. Even in nations with rule of law, United States applies strong pressure and registrars and server hosts give up bitcoin address used to pay for things.
They trace address to address until find person who sold bitcoins for cash or credit card or wire fee.
If cash, famous exchanges know which city and bank address cash was deposited. Modern law enforcement technique is excellent for figuring identity once given such a clue. If not famous and idiot from Craigslit or similar, roll over in hurry when threatened with prison. Will try very hard to remember. And credit card or wire more even more trivial to figure.
I run tests, bitcoin mining take 22 million million years to give pocket change. Computer not so fancy after all.
Forgive poverty english please, translated to many languages and back before posting.