What happens if a perfectly "healthy" bank falls victim to rumors and a bank run begins?

There have been several bank runs in countries with central banks, either for targeted banks or for entire banking systems. The most effective, if inconvenient, approach is the bank holiday. The central bank closes the banks for several days and reviews their books. Once that is done, the banks are either merged with stronger banks, bailed out, or reopen. By freezing everything in place during the bank holiday, the panic that is besetting depositors quiets down and people look at realistic concerns. Sometimes, as in Cyprus, there were real problems and shareholders where wiped out and large creditors, including large depositors, took a serious haircut. The problem with saying that a bank is solid is that it may be solid if there are no economic problems, but because they are so highly leveraged a small problem in the economy can become a huge problem for the bank. It is particularly problematic if the controlling shareholder of a bank has been using it to lend money to his other operations. In this particular case, it seems that the Bulgarian central bank should work with the ECB to quiet concerns. I don't know anything about deposit insurance in Bulgaria, but small depositors should be protected, no matter what happens.

From your question details it looks like the Central Bank is doing the right thing. Usually in these cases Central Banks impose blackout periods or "cool down" periods for unfounded panic to go away and to bring back reason into the game. During these periods, books are reviewed and quick stress tests are conducted, to assess the individual and systematic financial risks. Sometimes, cool down periods can be sufficient. But even with "serious" banks things can get ugly really quickly due to leverage, asset prices falling (because of selloffs) and reserves already low (because of withdrawals). The good side of leverage is that it is much more efficient to help and/or reorganize the stressed banks (even by "printing money") than to bail out depositors.

I'm not familiar with the Bulgarian system. In the US there have been bank failures due to large bank runs. Assuming the bank is healthy and how large the run is they should be able to recover by managing their balance sheet (deposit and loan portfolios). If the bank fails, or is in danger of doing so, then one of two things will happen:  they will be acquired by another bank that will manage it's assets, or if they are not acquired, the FDIC will pay depositors directly for the amount of their deposits they were insured. So, assuming that the loan portfolio is good (which you suggest) than if things get bad and the bank is going under, then there is a strong likelihood another financial institution will acquire the bank(s) in question. Or, your government might be willing to send/lend the institution money to assist with liquidity issues. If the bank is sound, other than the run, the government might see this as a safe and necessary way to keep the economy from declining (similar to what happened in 2008-09 in the US). However if the loan portfolio is bad then another bank may not want to acquire the assets. Then it's up to the government and their organization (FDIC in the US) to cover insured deposits. Deposits over the insured amount may not be paid and the customer may then lose money.

Bank runs should not be a problem in the USA.  The Federal Reserve exists primarily to prevent this problem.  Before creation of the fed, there were frequent panics where banks were attacked and ran out of money to pay depositors.  Today the fed would provide emergency cash to any member bank as needed to satisfy depositor demands for withdrawals. As we have seen for the last 6 years, there is no limit to the amount of money the fed can produce for this or any other purpose.  Through a program called quantitative easing it has produced trillions of dollars of previously nonexisting dollars and distributed these new dollars around the country in the hope of solving the problems created through the housing bubble.

First you got to understand that bank runs are problematic only because "healthy" banks also falls prey to them. banks that are not managed carefully or are piling up huge NPA's, are expected to fail eventually. if "healthy" banks not fall to banks runs then there is no need to worry about bank runs at all. Now, only thing that can safeguard a healthy bank from falling victim to bank runs is  confidence among public. bank runs are "self fulfilling", we expect a bank to go bank run , we withdraw all our money, hence bank actually goes bank run, fulfilling our own prophecy. I'm not aware of Bulgaria's banking regulations, but safety of all deposits in commercial banks are actually guaranteed by central bank. so if people believe in CB, then healthy banks will not suffer. So, Bulgaria's CB has to be credible enough. Credibility of course depends CB's past policies, and credibility of current top technocrats working in CB like Governor or chairman.