Many important statistical principles are best understood when they are seen to be violated. This creates a demand for nutty models.
I have not done the hard work that is needed to prove that the models given below are genuinely nutty, but I list them here to remind myself (and others) where one might look for "instructive examples."
John Tukey was remarkably able to avoid being ensnarled by complex models. He seemed to have a deep inner sense that even the simplest of things were complicated enough. Among Tukey's many inventions, one of my favorites is his notion of an antihubrisine. By analogy with an antihistamine, one should take an antihubrisine whenever threatened by statistical hubris, such as the thought that you know a correlation to within plus or minus 5%. Antihubrisine were introduced in Tukey's essay Sunset Salvo.