Once you buy bitcoin the coins is linked to address under your account of the exchange.  

And if you did send large amount to certain account.   You better have a good reason why you did it.   Money is still trace by the very first address.  No way to hide the money from that point.

If you mined the coins yourself.  It's possible to be anonymous.   As long you never link your IP address or your name to that specific mining address.  Have the client hide behind the VPN and tor.

If you buy the BTC in person with cash.   Both sides must not give out the real name during the transaction.   Huge disadvantage is you might give the cash to the wrong person even with username unless you find a way to secure the message without being intercepted.