One such blockchain that's giving it a good crack is DNotes...
DNotes (
dnotescoin.com) is cryptocurrency and payments system that acts very similarly to a bank, however with the added benefits of blockchain technology of course. The digital currency features a web-accessible fortress that securely stores user assets offline in cold storage, in order to avoid the associated online risks due to hackers, system crashes, etc. Additionally, all funds in the DNotesVault are covered by a 100% deposit guarantee, which is made possible through a separate cold storage account, secured in a different location. The secondary cold storage account will always hold an amount in excess of the Vault's total guaranteed deposits and are verifiable at any time through the blockchain.
While the funds are securely held in the DNotesVault they accrue interest, again very similarly to a bank but far more generous - with the Cryptocurrency Investment Savings Plan (CRISP). Each DNotes wallet will earn 0.5% interest every month, 6% annualized. This system rewards investors in-line with the network's long-term goals, with those who simply trade the currency forgoing this privilege.
The goal of DNotes is to provide a stable and reliable currency so that it can be used as you would traditional money, but faster, safer, cheaper and more transparently than ever before with no middlemen to slow down processes and rob its customers with devious fees (as banks do). Furthermore DNotes does not discriminate and is available to anyone, no matter where they may be in the world.
DNotes are building intrinsic value into the DNotes Coin with support from the strategic ecosystem that it is planned to develop, as well as 25% equity in the partnered for-profit DNotes Global Inc company. Furthermore, with a self governance POS model that rewards its own community for verifying the blockchain transactions, further protects the long-term interest and economy of the network - as those personally invested are less likely to dump newly generated coins.
So in summary to answer your question, yes - Blockchain can absolutely replace a bank, provided there is a comprehensive strategy to support the associated digital currency and that plan is effectively implemented and embraced. DNotes could potentially be just the blockchain to do it.