Unity web player is not installed correctly

I've installed the Unity web player multiple times on a PC system, but when I try to run a game, I am getting the following message:

Unity web player is not installed correctly. Some files are missing or corrupt in the Unity web player installations files. Please try to re-install.

I read some other threads about this, and followed the instructions many people are giving:

I uninstalled the web player, rebooted my system, downloaded the newest web player from the Unity3D website, installed it again, rebooted my PC, and tried to run the game again, but no results so far, except for the same error message.. On any other system that we have here, the web player runs normally.