Why do only some of these point lights work?

This is because for performance related reasons Forward Renderers limit number of light sources that are calculated per mesh.

Note: This limitations does not exists in precomputed light maps.

Immediate workaround is to increase number of lights per mesh in quality settings. But this is not a good solution but more of a temporary shortcut.

Solutions

  1. Fake these light with fx, glowing decals and lens flares. Most games do this.
  2. Consider switch to Deferred Renderer after familiarizing yourself with it’s performance and compatibility implications first.

possible duplicate: