With adequate processing power and network bandwidth, I think that it's possible for blockhain to someday replace banking. In fact, I even think that someday, cash will be completely obsolete leaving only digital currencies to remain. Blockchain makes lives easier a hundred folds and humans always go for the easier route. However, it may probably take another decade or so for that to happen.
Blockchain is one of the competitors of banks. Banks are attracted by the transparency of operations that all participants can watch. I think that the blockchain will not replace the banks, it will merge together with the banks.