No, agreements between the companies ensure that only one option is available to each region. My city can offer Time Warner the right to run cable here whenever they want, but Time Warner will refuse to do so, because Time Warner has a deal with Comcast that says they will never offer cable in my city. There is no legal barrier to new cable startups like Google Fiber, only the immense cost of it and the threat of retaliatory blacklisting and throttling from the cable monopolies.

That's why the only places these companies compete are ones where the cities have decided to set up a local public broadband to buck the monopoly, or a very well resourced competitor like Google has decided to enter the market and refuses to play their anticompetition game.