A bank failure is a financial disaster, and the most central banks can do is damage control. It is a very difficult call to make whether or not to try and save a failing bank, when to intervene and to what extent to intervene. Whatever decision is taken is normally criticised by some or other group of stakeholders.
In addition, central banks can usually not disclose the information upon which their decisions were based. However, even central banks that focus primarily on price stability sometimes face possible conflicts of objectives. The most obvious one is probably the potential conflict between maintaining price stability and financial stability.