Can the fly-by-wire system of a modern airliner handle an unstable condition?
Yes it can - to an extent, depending on:


amount of CG shift;
airspeed;
aircraft weight (actually Moment of Inertia);
maximum deflection rate of elevators/ailerons.


If the airframe has aerodynamic static stability, a disturbance in Angle of Attack Creates aerodynamic forces that return the aircraft attitude back to neutral. If the Centre of Gravity shifts aft of the neutral point, the airframe becomes statically unstable - the aerodynamic forces now want to amplify the disturbance. The rate gyro's in the fly-by-wire system sense that the attitude deviates from the commanded attitude, and deflect the elevators/ailerons such that the aircraft is brought back.

Image source

Amount of CG shift
The ability of the Flight Control System to correct for unstable equilibrium depends on the degree of instability: the curvature of the right cup in the picture above.

Airspeed
Deflection of the elevators and ailerons generates aerodynamic moments proportional to $V^2$. At high airspeeds, a small deflection results in a large moment about the CoG - the critical situation is low airspeed, which mostly occur close to the ground. At low airspeeds large deflections are required.

Aircraft weight
If the aircraft is at a high gross weight, and if this weight is distributed away from the CoG, larger control deflections are required to correct disturbances in AoA. A heavier bowling ball in above picture makes for higher required forces.

Maximum deflection rate of elevators/ailerons
The frequency response of the flight control systems is a function of how fast the control surfaces can be brought to the required position, and this is a function of hydraulic fluid supply rate: the added capacity of all the operational hydraulic pumps. 

The F-16 is designed to rapidly correct for any disturbance from the commanded attitude, in any flight situation. All systems are dimensioned such, that artificial stability can still be provided at the lowest airspeed and the highest MOI that the aircraft can encounter. The A320 is not designed with a flight critical FBW system - if all systems fail, the aircraft returns to a state with aerodynamic static stability. If the FBW system fails in an F-16, the pilot must eject.