It depends upon how low a probability failure you're willing to consider and how paranoid you are. For one thing, the U.S. National Command Authority could decide that GPS represents a threat to national security and disable or derate GPS temporarily or indefinitely over a limited or unlimited area.
Derating GPS wouldn't affect the time reference functionality. Turning off GPS entirely would seriously affect military aviation operations.
Not so: "Selective Availability (SA) is the deliberate introduction of error by either altering the precise timekeeping of GPS satellites or the position of the satellites in space, through the on-board software, thereby reducing both positioning and timing accuracy for civilian users." GPS accuracy is generally reduced by adding noise to the timing. Now you would have to derate GPS pretty significantly before timing accuracy would be significantly affected. But it's possible that some time references would refuse to lock on at all with sufficient derating. The affects of more extreme derating than SA are, at least to some extent, unknown.
Aviators try very, very hard not to trust their lives to GPS.
As opposed to LORAN ?
Generally, aviators don't like SPOFs. So they try very hard not to trust their life to any one thing. GPS is used in conjunction with VORs, pilotage (navigation by reference to fixed objects), and dead reckoning. GPS is used for instrument approaches, but only under extremely controlled conditions by very experienced pilots. A significant fraction of instrument training is how to cross-check instruments and detect failures. GPS approaches are individually approved by the FAA and factors such as runway lighting are critical. FAA approved GPS units must be used and one of the things these GPS units must do is monitor signal integrity (RAIM). From time to time, you will read FAA accident reports of people who attempted to perform GPS approaches with just a handheld GPS. DS