Simple free market rules would come into effect. Consumer (you and I) aren't happy with the product (Verizon). A competitor sees this as a vulnerability to gain valuable market shares of the highly sought after ISP market. Consumer is happy with competitors service, albeit a bit more expensive than today's rates (possibly, most likely in fact), but I absolutely believe the services would be much, much faster than is currently present in the majority of the United States. Verizon would either have to take the market loss and hope that their additional "fast lane" users will compensate for that, or they'll revert to some other business model.
No, there's no competition in many places, plus it's very difficult and expensive to start a new ISP, and there's also shady agreements between ISPs so they can have control over prices, maintain their monopoly and keep new players from entering the market.