If we speaking kinda teoritally.. well, I guess it is possible sometimes.. maybe after all bitcoins would be mined.. and price could be more predictable.. but I'm sure banks will not offer for you very good conditions...
They will earn money for themselves first of all..
But I think it is good if banks can accept bitcoin, and I think it's already done . In the Philippines PNB , LandBank accepts money from bitcoin it was bitcoin that is converted to money and transferred to banks, I think there's no difference in it. The money was transferred virtually without anyone going on the actual bank or money transfer, just click and it'll be sent. And also the bitcoin or money will be used by the bank and it'll be an advantage for them .