It would if it's overheating. The coolant will get pushed to the overflow bottle, and may be getting forced out the overflow. I still don't believe teh fans are the problem while driving, unless you sit at a lot of stoplights on your commute. The air flowing through the radiator as you drive should be enough to keep it cool (except on days where the temp exceeds 85F or so. Before you go crazy, replace the t-stat and re-bleed the cooling system and see what happens. Also check the ends of the intake where it meets the heads and block for coolant leakage. You may have one of the rare instances of intake gasket failure where the coolant leaks outside, not inside the engine. Also check the quick connect hose fittings going to the heater core for signs of leakage. A cooling system pressue test may help narrow down where your leak is if the t-stat/bleeding doesn't fix it.