Well, the problem is that the game actually only allows 2 players to have the same serial number while playing over a LAN and that when 2 players have no serial number, the game actually also considers them to have the same serial number.

To fix this you'll have to create the following key in the registry editor:
"Computer\HKEY_LOCAL_MACHINE\SOFTWARE\Westwood\Tiberian Sun" (or if you have a 64-bit operating system it's "Computer\HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Westwood\Tiberian Sun"). Then in the Tiberian Sun key, create a new string value and name it "Serial" and simply enter some random number for its value data. So as long as the value data of "Serial" isn't the same for more than 2 players, everything should work fine now.

I explained this with the assumption you've used the registry editor before; if you haven't I could explain it in more detail if you need me to.