What is the secret to making a low poly model with good lighting?

I want to make models for games and so I keep my models at a relatively low poly count; however I’m always stuck with one problem : When I stick lights in somewhere to illuminate the some part of the character’s face, there are often weird lighting situations where the lighting is not smooth in some parts and I see jagged black boxes that don’t really correspond to the model at all. I had already fixed the normals with Ctrl N, but it still feels like I’m missing something else? I rendered the models in Blender btw.

Screenshot or .blend file?

You might have double vertices or bad geometry.