You can be anonymous when exchanging bitoin for fiat. If you sell you bitcoin to someone in person (localbitcoins.com), then you can turn your bitcoin into fiat and vice versa. I think this would be hard to track back especially if the person you transacted with did not know your identity.

By trading BTC in person, you can obtain bitcoin relatively anonymously.

If you use any service such as Coinbase, any transaction and subsequent transaction are no longer anonymous.