FDIC has about ~225 billion to backstop failing banks. This is sufficient to backstop 1-2 failing banks, depending on the size. However, if a system breakdown happens, the FDIC won’t be able to safe most depositers. Only if the Fed print prints a ton of money, this would be possible.
Discussion
It looks like in the SVB case, a special type of loan arrangement does the heavy lifting of the "printing money" part