Can you be insured by more than one auto insurance company?

I am thinking of buying a car and currently I am under my dad's plan for his car. I want to have a different insurance company for my car. Would I need to go off of my dad's insurance policy for his car for this? I am thinking the only problem would occur would be if there was an incident that didn't involve the insured car itself but where my auto insurance would cover whatever it is (e.g. as a pedestrian I am hit by a car in a no fault state). Thanks