The grace period you get for getting your car registered in a new state is a grace period for not having your car registered in your current state, not a grace period for not having your car registered.

The new state won't prohibit you from registering your car in that state if it doesn't have a current registration in another state.  But if you get pulled over without a current registration in any state, you can still be given a ticket for not having a registration.

In most jurisdictions, an expired registration is a secondary offense and you can't be pulled over for that.  But if you're pulled over for a primary offense (speeding, running a red light, etc) then they can add on a secondary violation for the expired registration.