If a society attempts to save more, its aggregate expenditure will decrease causing the equilibrium and real output to also decrease. As real output and income have shrunk, the society will not be able to save more, but only the same amount (or may be even less).
The great depression is a vivid and regrettable example of the paradox of thrift. People tried to put more money aside because they were afraid. But, their very reluctance to spend caused income to decrease.