It depends on what the limiting factor is for the weight limit. If the tires are the limiting factor, then it is possible to carry more with higher rated tires.
Usually the tires are not the main or only limitation, though; it's some combination of the axle, the brakes, the suspension, the tires, the wheels, the overall vehicle frame strength, and possibly some other things I'm not thinking of at the moment. If it were as simple as using different tires, surely the manufacturers would put on the heavier-duty tires and advertise the larger carrying capacity (or, more likely, consume it with additional features and glitz).