technically you can. why? 2 words: Satoshi Nakamoto.
It's arguable that the things that Satoshi has never actually implicated his real life identity. His likely use of a VPN/Tor has probably made it much more difficult for people to trace him. His involvement in Bitcoin was primarily on the development and his use of obfuscation techniques probably helped his anonymity tremendously.I doubt that would be the use case of an average joe when they use Bitcoin.
You cannot say NO and then say it is NO only for the average Joe.
The answer should be ...YES, but you have to apply strict measures to be 100% anonymous. You are also looking at this from the angle of someone hosting a node and not the average Joe that learnt how to make a secure Paper wallet. 
Throw Paper wallets and Mixers together and you have a better chance to stay anonymous... until you use those bitcoins at a regulated exchange or a third party service where your real identity is linked to your payment or delivery of goods.