For example if you owned a bar and had some contest where the “final four” guys standing (after a theoreticaly drinking contest, lets not get into the insanity of the legal problems a drinking contest would bring, it’s just a fun example) received 50 bar dollars couldn’t you advertise for this contest in a way that would possibly achieve both the advertising effect you desire and avoid giving the NCAA any real legal grounds to take action? I mean, seriously if you say, “Be one of the final four standing tonight and win 50 bar dollars” could the NCAA really win?
And you don’t want to play golf with me! So, there! 
Yes, it was a silly example which was obvious.
Most companies with a substantial portfolio of trademarks have branding guidelines to determine what goes out under what name. Simple example - The Gap, Inc. has three different stables of brands: the “Old Navy” discount/ low cost line (oriented more for families, esp. families with children), the “Gap” line (oriented more towards teens and college students), and the “Banana Republic” line that targets more of the post-college to 30s set. Gap deliberately restricts what kinds of products get offered under each brand line. Similarly, Disney has “Disney” for kids films, “Touchstone” for the PG-13/ teen/ young adult fare, and “Miramax” for the rated R/ more adult fare. In the U.S., the Pepsi brand appears nowhere on Aquafina bottles. This is probably because Aquafina is targeted to the healthy set that eschews soft drinks. In India, on the other hand, Aquafina is clearly identified as a Pepsi product. This is because in India, the Pepsi mark serves as a guarantee that the water is clean and safe to drink. (It’s still catering to the healthy set, I suppose, but with different health considerations.)
The NCAA is no different. They’ve made an internal decision, for whatever reason, to restrict how they use their own trademarks. Perhaps they feel that if they let “Final Four” be used for all the tournaments, it may start them down the slippery slope to having “Final Four” become generic for semifinals in any sport. I don’t really know the thought process, but substantively, it’s no different than Disney releasing “Pulp Fiction” under the Miramax label rather than the “Disney” label.
As far as MartinHyde’s question, whether a use is confusingly similar or likely to dilute the NCAA’s trademarks is a highly fact-specific inquiry. The example cited might well pass muster, although the NCAA would have a reasonable argument that associating its marks with a drinking contest is not so far removed from associating it with porn. But context is key - “final four” is not being used in that example to identify the source of a product or service or to distinguish that product or service from other services. So it arguably doesn’t implicate any interest of the NCAA at all. It’s the commercial use to identify the source that trips trademark law.