Wouldn't be a pleasant ride, and even shocks wouldn't be able to absorb all that bouncing. Solid tires would be so rigid as to cause you to experience every bump and hump, and not in a gentle way.
Armored cars (Brink's truck) have solid tires to make them harder to rob. But for everyone else, they would weigh too much, requiring larger engines with more power to turn them. Plus the ride would be brutal.