I'm sure that the movement of the coins can be anonymous, if you take the right precautions (VPN, TOR browser, etc.) But eventually if you ever want to cash them out, you're going to lose your anonymity pretty fast.
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Exactly. In the end, either your coins come from an exchange (connected to an identity) or will go to one. The only exceptions are for miners (who can generate their own coins) and ultimate holders (hold forever on blockchain).
However, the services you mentioned (VPN, TOR, etc) are really helpful to conceal yourself. Using an HD wallet also helps conceal your exact balance (because your coins are held over multiple wallets).