Are you sure you would pull your money out if all the banks started charging -0.1% annual interest? If you have your money in your house there's a higher risk of theft, more than enough to counter the -0.1%. Add in the inconvenience of moving large amounts of money around by hand, not being able to write checks, and no automated bill pay, and I'm still keeping my money in the bank.
Yes, I'm sure. I certainly wouldn't stuff it in a mattress as cash, though. I'd start by finding a bank or credit union that didn't, as I'd be surprised if they all made that move at once. If that failed, I'd look towards accounts with a broker instead, or some other institution.
It seems to me that in this situation savings accounts would have the nominally negative rate, whereas checking accounts would have a nominal rate of zero. Since inflation affects the real value of both accounts equally, then the checking accounts would have a higher real rate.
Since both accounts are covered by FDIC insurance [0], it seems that the only difference (unless I am missing something) lies in the rate of return, thus ceteris paribus and the checking accounts have a better rate of return compared to savings accounts. Both have real negative rates, so the situation would be an active discouragement to hold the government-managed currency in any form.
Regarding cash holdings, you are right that they are probably a bad idea. However, if numerous banks could potentially experience a crisis beyond the intervention capabilities of the Fed and Federal government, then it would probably be more sensible to move the money into cash and store it in safety deposit boxes, possibly across multiple banks. This to me seems an unlikely scenario, but not necessarily impossible.
I am reading much of your premise to be related to checking accounts (check writing and automated bill pay). Most people don't carry a large balance there so a negative interest rate on those accounts will not really have a large impact.
Yet on savings accounts, or those accounts whose intent it is to be liquid yet remain relatively unused, could be impacted by this. Which at that point, again under your premise of robbery, a person is essentially wagering whether they would rather be guaranteed to be fleeced incrementally, or run the possibly of someday being taken for all of their savings. At that point I think you would see many people start to withdraw large amounts of savings, whether to be stored underneath a mattress or into bonds or some other investment, while still keeping an active checking account.
Since most people don't do this already despite losing money to inflation, I doubt there will be a significant change in behavior for most people when interest rates tip negative.
Don't underestimate perception. Fees/taxes/"negative savings interest" are much more visible than inflation. I'd be surprised if that didn't cause bank runs.
For my part, I already carefully manage how much I keep liquid versus invested, but anything equivalent to a fee would cause me to switch institutions or otherwise drastically change this strategy.
