As far as I'm aware FMC's are the main source for navigation. although they can use GPS information to cross check the location the FMC can also use VOR's and NDB's for the same purpose.
I also believe that in the case of any information not getting though it can run on its own until the cross checking kicks in again whereby if a GPS is unable to get a good fix because of the poor satellite reception it gets downgraded to a point when its too inacurrate.
The GPS standard has improved over the years but still I've seen the error go haywire when the satelittes are on the horizion or grouped to close together.
However will say I'm not aircrew and this answer is just based on what I'v learnt myself.