Depends. If you are insuring the car for physical damage (collision, theft, etc.) alone, no. Because you can’t ethically put yourself in a position to gain financially from someone else’s loss. For example, an insurer can’t agree to pay YOU for damage that happens to MY car, because you have no INSURABLE INTEREST—you lose nothing if my car gets wrecked.
But if you’re talking about liability, then it depends. Usually, yes, you can do it. One of a couple ways. If the car belongs to Mr. Snuffy, he can lend it to you for a brief period and you will have coverage IF his policy covers permissive users. That depends on the insurance company. If Mr. Snuffy says, “Here, use my car for an indefinite period.” Then there are a couple ways you can handle this. The BEST way that comes to mind would be to have a policy issued in the names of “Mr. Snuffy and Anaamika.” That way if something happens to the car, his interest is covered as any physical damage settlements will include him as a payee. If you accidentally run over a kid, you will be covered for liability. Additionally, should Mr. Snuffy be sued for negligent entrustment of the vehicle to you (he knew you were an habitual drunkard but loaned you the car anyway) he will be covered.
And now: insurance laws and coverage specifics vary from state to state & company to company, speak to a local insurance professional about the details pertaining to your particular situation, etc. etc.
And see sunfish
's exquisite bit of research.