Answer:
Yes. Let's look at an example.
A 12 volt system can operate a 12 volt relay coil to actuate the plunger. But when the field of the coil collapses in the instant following when the coil is de-energized, the collapsing field can generate an "inductive kick" that can easily reach several hundred volts.
Relay coils frequently have diodes across their coils to shunt this voltage through the coil and prevent it reaching other parts of the circuit because it is a higher voltage and could damage components.
As a closing note, you're not "getting something for nothing" or generating more energy out of the coil than originally went in. The power in still equals the power out (less any loss). With a properly designed coil utilizing Marko Rodin's configuration, losses are negligible.