Fans operating in the atmosphere (not against a fixed downstream pressure) follow the cubed relationship well.
A variable-displacement swash plate pump in a hydraulic power unit (very common application) won't follow the cubed relationship, assuming the demand is set based on a fixed pressure setpoint (which is the normal case). Slowing it down while remaining within system capacity will just make the displacement-modulating mechanism go to a bigger displacement thus demanding the same (ish) power, i.e. more torque at lower RPM. Depending on where you are on the motor's torque curve, the motor might not be too happy about that, and it won't save power. Of course, if the pump displacement mechanism hits max pump displacement, the pump will no longer keep up with demand ... you'll use less power, by operating the load slower. Probably someone isn't going to be happy about that, either.
A centrifugal water pump operating against a fixed head won't follow the cubed relationship, either. Slow it down too much, and it simply stops pumping, acting dead-headed. That's not good.
Certainly there are cases where such applications can be slowed down (e.g. because it was oversized in the first place), but be prepared for the results.