@Wally: not according to boeing. An excerpt from our FCOM:
FMC logic selects the position from one of the GPS sensor units as the primary update to the FMC position. When GPS position data is available, radio updating can also occur. If all GPS data becomes unavailable, the FMC position will be determined by radio or inertial (IRS) updating.
New procedures like RNP AR approaches actually require us to disable VOR/VOR updating.