Bitcoin _can_ solve this problem because it is totally distributed and doesn't depend on traditional banks which means transferring value directly between individuals and organization becomes much simpler.
It is expected some Bitcoin users (be it private or institutional) will become quite bank-alike in some future (as per various posts on blogs and HN in discussions of Bitcoin).
In short, Bitcoin doesn't remove banks -- neither need for them, nor ability to operate them. They just haven't moved into this space yet.
EDIT:
first episode of ``The Ascent of Money'' [1], `Dreams of avarice', explains how banking, as we know it, came to be.
Yes, it does in this case. One can receive money electronically through Bitcoin, without account with a bank, PayPal or Google.
Since the Bitcoin protocol allows third-parties to hold user's wallets, it's entirely possible to create a bank. In fact, the protocol seems to encourage the aggregation of wallets.
The point is that you're moving your currency(BTC, USD) from you to another party without any intermediary interfering with the transaction. If you mail $1 to someone, that's effectively the same thing as paying them with 1 BTC.
EDIT: I think the people who are saying that Bitcoin cannot have banks are conflating payment processors with banks. A bank is a place where you go to put your money, and they use that money to make loans, which gets them more money through payments on loan interest. They incentivize this behavior(giving them money) by giving out interest on the amount in the accounts.
Payment processors are companies(i.e. Visa, MasterCard, etc.) that let you take your money from an account, and give it to someone else. Banks may offer these in the form of checking accounts, but that is not their intended purpose. These, I agree, are definitely not needed with the Bitcoin system, because that's what the entire protocol handles.