First, anyone can use Bitcoin reliably without a bank account. In fact, there is no user registration, no credit check, and certainly no central institution that can reliably exclude users. Completely detaching your life from banking, however, should still be somewhat challenging:
- To pay for groceries/clothing/gasoline/public transportation and other regular needs with Bitcoin will probably be hard. It can be done, but at this stage will require much more effort.
- Emergency situations would really want you to have a fall-back. A broke down car in the middle of nowhere and stuck trying to get the tower to accept Bitcoin would suck, or, well, needing a taxi ride to the ER.
- Obtaining bitcoins is harder without a bank account. You'd have to arrange to trade locally, which costs more time, is not without security risks, and a bit more expensive. Or convince your employer/your costumers to pay you in Bitcoin.
- And you'd need to find an internet service provider that allows you to pay in another fashion than anything banking related.
Bitcoin is private in that nobody can easily see how much money you are holding or spending. It is not anonymous, because whenever you trade Bitcoin for anything else, your identity might be breached to at least your business partner, thus create a connection to your address or even several of your addresses.
With some nations having already declared Bitcoin's legal status (e.g. Germany), and some online services introducing Bitcoin as payment option(e.g. Shopify), also judging from news today (e.g. Time reporter doing her Christmas shopping exclusively with bitcoins), Bitcoin is spreading quickly. I'd say that with luck and a bit of patience, it might actually be comfortably possible to completely do without a bank account.