Gould and Eldridge found that many species are remarkably stable through long stretches of geologic time. I've seen their reports (Gould's papers are largely available free online, with a quick Google search, if you want to see them yourself), and even got the chance to look at some of the species they worked with, and can attest to the accuracy of their observations. I've also seen it demonstrated in species neither considered, which further supports this notion.
The reason for this, according to those researchers and later ones (and from my own research, I can say confidently that it makes sense), is that organisms will quickly adapt to reach a local fitness high. Once there, the organism can't easily adapt any further--they're at the local high, so any adaptation will reduce their fitness. There are numerous mechanisms by which this can happen, but natural selection is constantly culling the population, creating selection pressure that attempts to push the organisms back to that local fitness high. In other words, organisms rapidly adapt to their environment, and most of the time any subsequent shift in the population will result in the organisms being less fit.
This isn't universally true. Chronospecies are species that evolve wholesale into new species, and this obviously can't happen if the population is under confining pressure. And if the environment shifts significantly the organisms adapt, follow the environment, or die. But for many organisms, natural selection is a conservative force for most of the lifespan of the species.
The real issue--one that people on all sides of the PE debate tended to forget--is that biology is incredibly complex. PE, phyletic gradualism, and other modes of evolution--even hopeful monsters--play a role. The question is, WHAT role? In other words, which modes of evolution are dominant under what conditions? PE works well as a model for mollusk evolution, but not so well for foraminifera (or, as I said before, chronospecies). Hopeful monsters do from time to time arise (six-legged frogs and the like), but they tend to not survive very long and are almost irrelevant to evolution today (in the Tonian, however, who konws?). Plants and bacteria and archea play by their own rules; these models of evolution (and pretty much any you'll ever see on this forum) were developed for animals, so there's no reason to assume they apply. Etc.