Have you ever lost the budget for a new employee because the position was left unfilled for too long?

Yes...in bigger companies this is not uncommon, both for people, spend and capital. The particularly nasty thing about it is that everyone figures out this game really quickly, so GMs are incented to spend their full budgets under the "use it or lose it" theory. Cash savings, efficiency, etc can become a bad thing. I've seen awful employees hired/retained, budgets flushed for no purpose, just so the $ are "saved" for the next cycle.

The flip side of "why this is a good policy" is that it normalizes budgets if a GM (or their finance support) forecast badly or cannot execute, and that budget is moved elsewhere to someone who can (or to a biz unit that is simply being successful). It helps actuals adhere to forecast better...because everyone likes predictability this is a good thing.

One train of thought on this "if they've done with out and still performed perhaps they really didn't need the FTE"; another solution for the budget could be to job enrich existing employees with talent to take on these new or additional responsibilities and hire a more junior role at a lesser cost;