PROBLEM SOLVED!
The difference between the two models was that the one on the right had only one resolution LOD (it is still under development).
I did alot of manipulation, changing the LOD numbers, adding LODs , etc. The bottom line is: IF YOU HAVE ONLY ONE LOD, YOU WILL HAVE TROUBLE.
I knew this already, of course, because no modeler in his right mind would only make one resolution LOD. WHat I didn't know is that the model would behave oddly if it only had one LOD.
So, aspiring modelers and those of you who are experienced but did not know... If you are going to test your model, make a second resolution LOD even if it is an exact copy. Then you won't have this problem.