Cooked potatoes that are unrefrigerated will last for 2 to 4 hours before going bad.
If you want to keep the cooked potatoes longer than 4 hours you should refrigerate them.
Cooked potatoes that are properly stored in the refrigerator will last up to 5 days and sometimes a bit longer although after 5 days the cooked potatoes will begin getting hard and losing their freshness even though they might still be good.
If the cooked potatoes are left out at room temperature the bacteria will begin growing soon after 2 hours so it's best to not keep the cooked potatoes out of the refrigerator at room temperature longer than 2 hours.
Refrigerate the leftover cooked potatoes if not using them or eating them within 2 hours to prevent bacteria from growing on them.