So what actually happened is that Clinton said we had to get serious about the deficit, and then passed a budget that pushed the real cuts until after the end of Clinton's second term (that is, more than six years down the road).

Then Republicans took over the House in the midterms. They passed the budget (and Clinton signed it) that actually led to surplus for the first time since Andrew Jackson.

So, yes, it happened under a Democratic president. It only happened because of a Republican House, though.

(But let the president also be a Republican, and Republicans in Congress lose all fiscal discipline.)