Can a four-engine aircraft with limited fuel gain extra range by shutting down one of its engines?
Assuming the aircraft is always flying at its optimal Lift/Drag ratio (= optimal angle of attack = optimal speed for a given weight and altitude), then the question is:

Is it more efficient to generate a given amount of thrust T with 4 engines x T/4, or 3 engines x T/3?

I would think that in the general case, it is not the case. Specific Fuel Consumption tends to increase with thrust (at constant speed):



Note: the above diagram is for Turbofans (Source: Airplane Aerodynamics and Performance, Roskam). It would be interesting to have similar trends for other types of engines (turboprops, turbojets, piston engines), but I could not find any!

On the other hand, it is possible that, in this particular case, the engines were simply working too far off their design point, so increasing thrust was improving their SFC.

Also, there is a problem with cutting one engine: The pilot will need to counteract the thrust asymmetry with the rudder, and this will have a significant cost in drag. Of course, the pilot could just cut 2 engines instead of one! :-)