I'm not disputing whether there is an ill-conditioning present here; I am saying that it is not the system that is ill-conditioned or otherwise somehow particularly pathological, and that this is obviously not due to a modeling error. Please consider re-reading my comments since my point is something other than what you are disputing -- you seem to be disputing a point that I was never making.

It is not the system that is ill-conditioned or pathological, nor is it modelling error, I agree. It is the comparison x(t) >= h that is ill-conditioned, and you'd be wrong to rely on its result (much like catastrophic cancellation, it is a bad idea to compare approximately equal numbers with errors in them, no matter where the errors come from—the errors could come from measurements, and the same logic would apply). We might have different ideas about what in your example is specified as given (with full precision) and what is specified approximately? But I do not think I misunderstood anything you wrote, please consider re-reading my comments too.