Bitcoin is anonymous to a degree.
You do not have to use your credit or debit cards to purchase Bitcoin, you can complete cash transactions via meet ups or Bitcoin ATM's and at that point there is no tie to your bank accounts. However, Bitcoin is really only pseudonymous. You use an alter ego, or a pseudonym, in order to complete transactions and move BTC. However, every coin carries with it a history as it moves through the blockchain. Sooner or later, if needed, that coins history can be traced back to certain accounts and transactions which are called into question. There are other coins that use a technique called "mixing" which essentially erases each coins history, and those coins are anonymous.
So, in conclusion, yes, Bitcoin is anonymous, but be careful, Bitcoin's level of anonymity is by no means invincibility.