No, they couldn't. One person's savings is another person's debt. If everyone tries to save then no one can; this is the paradox of thrift.
Just because someone gives it a formal name doesn't mean it's true.
>One person's savings is another person's debt.
Only if you treat fiat currency (or bank deposits in general, the paradox you mention was formulated long before we got off the gold standard) as the only possible form of savings.