Lighting problem

I have a problem where some of the parts of my scene don’t seem to be lighting properly - they jump from black to white when the light touches them and jump back to black as soon as the light leaves them. Other parts of the scene with the same model and material work properly. The objects in question use the built-in Diffuse material.

Here’s a video of the problem in action

Looks like a forward lighting problem, where you have more lights hitting the objects than allowed in the Quality Settings.

Try to increase the Pixel Light Count setting in the Project’s Settings and see if that helps.
If not, try switching the rendering path to deferred lighting in the Player Settings.