Obviously if there is "a run" on a bank's cash reserve, their obviously most common solution is to BORROW cash at interest from another bank. Sometimes this money comes from a central bank, but normally it comes from one of the other banks who are not experiencing "a run" on their cash.
A bank does not "go bankrupt" when it runs out of cash. It goes bankrupt when it runs out of credit worthiness. Banking is always based upon trust in the bankers making the decisions.
A bank that fails to meet deposits upon demand is said to be "in default", not "bankrupt". A minor technical difference that to ordinary depositors doesn't really make any difference either. There are also the additional problems of theft and embezzlement that can put obligations as higher than assets (money owed the bank through loans). At that point the bank really is bankrupt and out of cash no matter how much comes in from the loans. Borrowing money from another bank isn't going to help.
A huge problem with Bitcoins is that there is no "central bank" to run off to in terms of acquiring a quick loan between banks. There may be some banks that are larger in terms of Bitcoins accounts and better run, but a central bank in the traditional sense like the Bank of England or the U.S. Federal Reserve simply can't exist for Bitcoins. The very nature of Bitcoins precludes any potential bank of last resort, or somebody deliberately debasing the currency with a huge influx of new money created out of nothing. Yes, Bitcoins does have a coin generation system, but that is very egalitarian in nature and random in terms of who will get that money.