That's taking it into extremes. We don't need to go all the way with this. Make sure most of your money is secured somewhere like in a bank deposit box, since those aren't counted into bank's liabilities and as far as I know cannot be taken away in case of bankruptcy.
I have some money on my account. It's less than 10% of all that I have, but I do it because it's convenient. If I had all my money at home in a vault, every time there's a bill to pay I'd have to drive to a nearest post office to pay it. The current system doesn't give you many options. You can say you won't use the bank but you'll either have to use another service like paypal or pay additional fees at the post office and waste your time and money getting there.