> How can any organization that issues its own fiat currency become insolvent?

What is the difference between insolvent and still existing but nobody trust / wants your currency? The US can issue currency all it wants, but if nobody wants it that does no good. My boss (I live in the US) could pay me in Euros, Rupees, or any of a number of other currencies - we already have plenty of employees who get their pay in the above currencies. I have to pay my taxes in US dollars, and the local stores all only take dollars, so I want dollars. However if things get bad I'll take some other currency and trade only for dollars as I need them - there is no reason my local grocery store couldn't take a different currency. If they want that currency they will give me a discount for using it.

We know from other countries where this happens, taking other currency is done on the black market. The stores and my pay is officially in the local currency, but everybody finds a way to ensure that they do as much as possible in a different system. (What really happens is far more complex than the above and not something I understand well enough to put into a comment, thus the gross oversimplification that is false because of it)