Bank runs and panics are not necessarily the same things. Bank runs were typically caused when people were concerned that a specific bank did not have enough "species" (gold or silver coinage) to cover their bank notes which was passed as currency. Panics were credit contractions where continuing credit was refused and existing loans called (often orchestrated to seize the assets of borrowers)which also caused contraction in the money supply.
No, the Fed did not solve the problem. Today we all know and accept that our banks do not have enough "assets" to cover your deposits. (In fact your deposit is not an asset - it is a loan to the bank which the bank can default on and comes behind the other banks, and even derivatives obligations in a bank bankruptcy) But bank runs do occur when people are afraid that a bank will fail and default on their loan from you. There have also been more credit contractions and panics in the 100 years since the Fed was chartered than the 100 years before.
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