The NHL, for example, first went to a 16-team playoff right around the time the now-defunct WHA was absorbed into the league beginning in the 1979-80 season. At the time, there were only 21 NHL teams, so only 5 missed the playoffs. There has been no expansion of the playoffs since then, but the NHL now has 30 teams, so 14 miss the playoffs. The upside is that that development makes the regular season a little more important. Same thing with the NBA -- the playoffs expanded to 16 teams back when there were still only 23 NBA teams. Today there are 30 and still a 16-team playoff field, which makes the regular season a little more important than it used to be.
MLB has seen a rather dramatic playoff expansion, going from 4 teams as recently as 1993 (the last season prior to the strike-shortened season) to 10 teams today. But MLB was so exclusive in its playoffs that the number itself doesn't seem terribly high to me, and the wildcard round is single elimination. But playing the World Series into November still sits a little odd with me.
The larger issue is the same at both the college and pro sports level: bluntly put, it's all about a money grab. In pro sports, there are arguably more cities capable of supporting a major league sports franchise than there are available franchises. So owners will threaten to move a franchise in order to extort more money from its present city. The leagues have hit on expansion as one possible way to protect the fans in the most vulnerable cities. As for colleges, larger conferences are capable of demanding larger television fees than are smaller conferences, and often the expansion results in a larger per capita fee for the pre-existing members. During the Big Ten expansion talks, Frank the Tank described that as "11 + 1 = 13."