First of all I dont think so that this will ever happen. Bitcoin being decentralized aims at eliminating intermediaries involved in financial transactions. This makes banks totally useless. Why would banks be so welcoming and friendly towards Bitcoin under these situations? Second, even if it accepts such deposits, it would not be very attractive. I mean I will not deposit my Bitcoins there as I can earn much from the market than a small, stable rate provided by banks.
If we have a decentralized currency and we have decentralized exchanges how long it will take until we have a decentralized platform to lend money with bitcoin? This going to be something complex but it will happen and when that happens all the possible reasons of why we could need a bank are going to disappear and we could stop using banks in a definite way.