Besides that, though, the entire mechanism of Bitcoin is one where all transactions are recorded publicly in perpetuity. It's really the polar opposite of private in that respect. It means that if your identity is ever linked to a particular bitcoin address (as in the not-so-unlikely scenario that the authorities or some other attacker are monitoring bitcoin-for-cash machines), then all transactions linked to that address can be linked to you.
In short, as always, it depends on your threat model. Are you trying to hide the fact that you forwarded sensitive nuclear secrets to overseas actors from a determined U.S. Government investigation? Sorry, you probably can't, at least without a bit of luck. Are you trying to fool Netflix long enough to stream latest season of Arrested Development? Probably doable.