The only instance in which government surpluses do not raise national saving is the so-called Ricardian equivalence scenario. In this scenario, households learn that the government plans to run a surplus, and they reduce their own saving by their proportionate share of this surplus.
In general, economists find that if Ricardian offsets exist at all, they are a modest share of the relevant government surpluses. At a more casual level, I have, for many years surveyed my non-economist friends to see how they alter their own consumption behavior in response to national fiscal policy.