A bank called Skandiabanken in Norway now supports Bitcoin. However you cannot purchase the Bitcoins through the bank, you'll be redirected to Coinbase in order to buy.
I don't have this bank myself, so I cannot test this. It seems like it's just linked to Coinbase and the bank does not store the coins on their servers.
This feature is in a beta phase, and the laws in Norway could cause conflicts. I think this is a huge step, and I hope this will soon be supported in more banks. This feature is mainly for investors, you cannot use it to pay bills and such.
This just sound a bit, meh. Local exchanges in our country allow us to buy bitcoins using our money from most banks here. This seem to be the setup with this bank you mentioned. What I'm curious is, is it the only bank there that allow this?