See clip from Attorney Rajiv S. Khanna's conference call video that addresses this question.
The question here is can I have permanent residency in more than one country?
Yes. You can. I can only comment about USA. I don't know about other countries, so if you have a Green Card in the United States there are possibilities you have a Green Card in U.S. and you have permanent residence for Canada as well. But in fact you stay in USA permanently, you work here, you stay here we really don't care what other permanent residency you have. So if as a matter of fact you are living in the United States we don't care how many other permanent residency you have.
If on the other hand, you quit your job, go to other country and take permanent residency there, it can be an issue for your Green Card here in the United States. Government can ask you, what is your intention. I would advice you to take detail consultation with your lawyer and make sure your particular circumstances covered. I think re-entry can be applied to protect your Green Card. But in that case getting a permanent residency in a third country may not be very good idea. However temporary visa would be OK specially when combined with re-entry permit, Form I-131 and or N-470 to preserve your Naturalization.
So look into that before you make any firm plan. As a theoretical matter if you are permanent resident of USA and you are living here we don't care how many other permanent residency you have.