Depends on the kind of bag and where it was purchased. Weight is another matter. But I've been places where I've held onto 4 heavy-duty plastic bags with handles weighing maybe 30 lbs total. It was just to the car, but I think I could handle that a couple of miles if I had to.
I remember years ago my mom thought we might just walk to a supermarket about 4 blocks from home and we walked back with groceries in kraft paper bags. The traditional kind without any handles. Those had limits - one per arm.