Yes, they did sit around bank vaults for years, but not the Federal reserve, since there wasn't one. Since the US stopped coining gold and silver dollars in 1804, these were the largest denomination coins in the country. Most of them sat in bank vaults as reserves and collateral for paper money issues. This is the reason bust halves are common in higher grades, and other coins like quarters are not. By the time numismatics started in this country, there still were many of these halves in bank vaults. A lot of early numismatists were able to acquire uncirculated examples this way. The rest were released into circulation, but since they were old many were then saved by the public after just light to moderate wear. Just try to find a nice AU bust quarter and you will see how much rarer they are than the halves. This is why.
Thanks Medoraman, that is very informative. I was actually just looking into some bust halfs and wondering why they were priced as low as they are given their mintage, seems like their status as reserves increased their supply.