Theoretically a new company could try to compete but the major companies tend to choose not to start price wars and it's a very expensive proposition to run competing cable out to houses, particularly since the incumbent will usually lower prices to make it more expensive for the new entrant. Around here, in the areas which have competition both Verizon and Comcast will cut and remove the other company's cables when they do a new install to maximize the cost and inconvenience of switching providers.
I have no idea where you heard this fairy tale. Comcast and Verizon sub out almost all their installs and line work to local contractors -- and most of these contractors have contracts with multiple telcos and cable companies. If the contractors are maliciously cutting wires, it's to get more work for themselves.
I have both DSL and cable modem at my house. There was no cutting of the first when the second was installed. The installer has no way to tell if that phone line is running DSL or if I'm still getting cable TV over that coax. That would not be tolerated for a second.
4 friends and coworkers who had it happen, mentioned separately and on several occasions this happened in a group setting and was immediately confirmed by other people who had the same experience.
You'll forgive me for finding them more convincing than your argument from incredulity. It's entirely possible that this a grassroots thing but the people affected spanned ~100 miles and multiple states.
http://usatoday30.usatoday.com/tech/products/services/2007-0...
http://www.cnet.com/news/cutting-the-copper-on-fios/
http://www.dcurbanmom.com/jforum/posts/list/352765.page
Note, however, that there also do appear to be a significant number of anecdotal cases of 'vandalism' that are exactly as described (competitors getting their lines cut), especially in the case of multi-unit dwellings with different subs.