Usually not. However, usually you get basic liability coverage as part of your rental. And if you have a good selection of credit cards, check into what protection is offered under those when you rent a car with them. For instance, the Chase Sapphire Preferred offers primary coverage for all rental cars, up to the full replacement cost, for those booked on the card. There are others with similar perks.
Also remember that insurance is the major add on purchase sold to your by rental agencies. They are most definitely trying to sell you something, and any insurance advice you get from them is for their benefit, not yours. So (as you appear to be doing) do your own research and don't trust the agency. :)
