Most won't.
While their bill would go up, switching means buying another phone and actually putting out effort, something most people really won't do. Most people are too lazy to even research a cell service. Which is exactly why AT&T and Verizon can charge such insane amounts.
Not to mention spending $100 on another phone just to save $5 a month means you aren't exactly going to be saving money for quite some time. 20 months just to break even, what a deal!
And I wouldn't count on better service. There is one neighborhood a few miles from me where the only service you can get is T-Mobile cellular service (no data), and usually only while outdoors. In my own neighborhood every carrier works indoors and out for all services, except T-Mobile. You can sometimes get calls indoors, but you can forget data entirely.
It just depends on your location as to how good service will be.