If the question is are you allowed to use two different sets of plates at once then, the answer is absolutely not. Not legally, anyway.
By law you have to register your vehicle.
Your vehicle registration has the state in which it is registered and your license plate number, as well as the VIN. That is what differentiates that vehicle from all others. Ultimately the purpose of it all is to protect you should your car ever get stolen.

 

keep them personally? maybe, some places just tell you to destroy them, but there is nothing forcing you to do that. some places will make you turn them in before they will give you new ones. Not sure what Nevada does so sorry, not much help

 

Also, which ever's state car insurance is lower, keep those plates registered. Regardless of what state you're currently living in. Trust me.