From the “Collecting Paper” column in the Nov. 21, 2016, issue of Coin World:

Paper money acceptance has historically been based on the fact that it was backed by something other than “the full faith and credit” of the issuer.

Because paper really has no value in and of itself, the willingness to accept it was often based on the belief that you could ask for the equivalent of its face value in hard money; usually silver or gold coins. 

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In antebellum America, it was generally assumed that bank notes would be converted on demand into coin of some type. Before enactment of the Coinage Act of 1857, that could include a variety of world coins that were accepted as legal tender in the United States. 

So when was this an issue? Well generally, in times of economic recession or depression, people started to worry about how safe their money was; remember this was a century or so before the advent of the Federal Deposit Insurance Corporation and other federal protection plans, and was also a function of the fact that the issuer of the paper money was not the federal government but typically a local bank that might or might not be a pillar of financial strength. 

During the Panics of 1837 and 1857 and, to a lesser extent, during the recession of 1854 and earlier periods of financial retrenchment after the War of 1812, nervousness led to concern, which led to panic, which led to efforts to get something tangible for your paper money before it was “too late.” 

While note holders and depositors milled around in long lines outside of a bank waiting to get their money, banks had a couple of strategies that could be followed:
1. “Lock the Doors” — this was done if the bank knew that the coins it had on hand had no chance of satisfying demands placed upon the bank for redemptions. This only delayed the inevitable, but might work if the city’s or state’s banks suspended specie payments en masse as happened several times.
2. “Buy Time” — if the bank was well respected, tellers might be instructed to pay off note holders in the smallest silver coin available, the half dime.As you can imagine, paying off even $50 in notes with half dimes that were very carefully and very slowly counted out by experienced tellers took forever.  

In the time it took to count out all the dimes, so the theory went, people would calm down, decide that the world as they knew it was not ending, and decide that maybe everything was OK after all. Really clever bankers sometimes planted a “customer” in the line that, with great flair, deposited money. 

The downside for the banks was that, typically, refusing to redeem even one note in specie on demand was grounds for suspension and forfeiture of their charters. However, during widespread panics such as in 1837 and 1857, sometimes this enforcement was waived temporarily.