Wouldn't happen. The insurance companies would sue it to death

As far as I know, and someone correct me if I'm wrong, but policies are written per driver and per vehicle.
If I were not to use my car to travel around between sites at work, but a "company vehicle" they would have to get out a policy to cover me on all vehicles registered to them (or at least the one or two they'd let me use).
So I doubt this is likely to happen because of being too cost prohibitive.