That's what banks choose to do. People depositing their money with a bank don't have a responsibility to ensure it remains solvent though.
If you think there is a significant risk your bank is going to go under, you get your money out. If you think that risk is due to panic about a bank run kicked off by an unfounded rumor, you get your money out. You take care of your affairs, and the bank can take care of its operations and public relations.