What phenomenon are at play such that the "Big Four" are limited to four and not a growing list? I've heard a list of anime referred to as "The Big Four":


Dragon Ball Z
Naruto/Naruto Shippuuden
One Piece
Bleach


What phenomenon are at play that there aren't more than 4 anime like these?  

I think the criteria to get into The Big Four are along these lines:


based on manga
long running anime (hundreds of episodes)
longer or concurrently running manga
possibly currently running anime (they got their name when Bleach was still airing, so 3/4 were running for a while)


I've seen other shounen series pass 100 episodes, and there are certainly other fan favorites.  There are also really long-running series geared towards children.  However, it seems that most anime nowadays is 12 episodes, and maybe a followup season of another 12 episodes.  I can't help but think that these phenomena are related.
I've heard a list of anime referred to as "The Big Four":


The first thing to be aware of is this: the idea of the "Big Four" is localized to the English-speaking / Western (possibly just North American?) fanbase. There is not an equivalent notion in Japan. This strongly suggests that an explanation for the "Big Four" must involve an understanding of the habits of English-speaking fans.


  What phenomenon are at play that there aren't more than 4 anime like these?


I tentatively am inclined to attribute this idea to the fact that the current crop of teenagers and young adults who are the primary inhabitants of English-language anime discussion websites grew up watching these four shows (in, say, the late '90s and early '00s). All four of these shows had a number of properties that were instrumental to their having broad popularity among the demographic in question. In particular:


they were dubbed in English
they were extensively aired on readily-available television channels (especially Toonami)
they were not "kid's shows" in the same way that something like Pokemon or Digimon is
they had multiple hundreds of episodes


The only other show that comes to mind with these properties is Sailor Moon, and for simple demographic reasons, it's unlikely that fans of the "Big Four" would be fans of Sailor Moon, and vice versa.

So why is this list unchanging? Well, I expect that the list will change in a few years, when more older viewers phase out of online discussion sites and more younger viewers phase in. (This is a simple consequence of the demographics of anime discussion on the internet - older people get lives and stop watching/discussing anime, and new youngfolk with nothing better to do figure out how to internet and start posting about things in their stead.)

These younger viewers will likely have less exposure to Dragon Ball Z (and perhaps Bleach as well) and more exposure to the newest show that has all those properties: Fairy Tail. Perhaps there will be a "New Big Three" in 2020, consisting of Fairy Tail, Naruto, and One Piece.


  However, it seems that most anime nowadays is 12 episodes, and maybe a followup season of another episodes. 


This was already the case by the time One Piece started airing in 1999 (and even earlier than that), roughly speaking. Forever-running shows like the "Big Four" have long been in the minority. You, the English-speaking viewer, likely just didn't notice this during your younger days when you watched the "Big Four" on television, since these shorter, one-to-two cour shows were less likely to be dubbed and shown on television (and if they were, they probably wouldn't be shown during cartoon primetime).



I should add that I am more aware of the idea of the "Big Three" (those four minus Dragon Ball Z) being a thing in the zeitgeist; and indeed, I suspect this is probably because Dragon Ball Z finished up a long time ago, and hence is not in vogue with the current crop of people who discuss anime online. (Whereas: Naruto and One Piece are still running, and Bleach ended only relatively recently.)



Random832 also points out in a comment that one article from 2012 claims that the "Big Three" consists of Bleach, Naruto, and Inuyasha. This brings to mind an important point: there is no "official" notion of a "Big Three" in anime - I'm not aware of any industry sources that use the term. The "Big n" is what the internet makes of it; no more, no less.