The GPS coordinate system is a cartesian system, centered at the center of earth. Thus there's only one possible position for each coordinate. The map datum used will make the projection on the earth's surface different, but not around half the world.
When using under-determined solutions, there may be alternate positions that are mathematically possible, but they are usually moving at unrealistic speeds or at unrealistic elevations (space travel), so they are pretty easy to detect.
Today's GPS receivers are always multi-channel (twelve or so) and normally runs with over-determined solutions. Thus they rarely get confused because of loosing contact with a satellite or two.
Still, I've had a Garmin eTrex Vista travelling across the North Sea at about 250000 km/h for half a minute or so, until it suddenly locked up and restarted, so everything can go wrong.