It's true, but not always!
If the whole economy are down and there's a big problem for all country in the world e.g. virus, war etc.
FDIC will not able to recover your money because they're also struggling!
Also, FDIC has limits on the mount they could recover for private person or company investment in bank. It differs from country to country, but if a persons holds a million in bank, it does not mean that in case of banks bankrupt he will get that million back. For example in Russia in case bank goes bankrupt, a physical person can only get around $17k back. In current realities that is nothing. And compensation isnt instant. Also banks can not just go bankrupt, but lose license. Persons funds are also get frozen, and he gets zero back. Banks are safety box.