The problem is that when there's the pitot-static failure the VSI (generally) indicates by "sticking" wherever it was when the failure occurred.

"Speed" means a different thing to a pilot than it does to a non-pilot. Groundspeed is irrelevant (aerodynamically), airspeed is everything. To sense "overspeed" you have to know airspeed. And since airspeed is a measure of the movement of the aircraft relative to the medium it's moving through (air), there has to be a system to measure that movement. That system is the pitot-static system. The pitot tube sticks into the uninterrupted airflow, usually on the wing somewhere, the static port sits where the air is calm, usually on the fuselage. The airspeed indicator in particular relies on the differential between the two inputs to determine airspeed.

Here's the best non-pilot friendly article I could find that explains what happens when the pitot-static system fails: http://www.boldmethod.com/learn-to-fly/systems/airspeed-indi...

I'm a former pilot, PPL with a few IFR lessons, and I'm pretty rusty on this stuff so while I'm pretty sure I'm accurate here YMMV. :)