Bottom line, the more I learn about statistics, the less I trust published papers in my field; I simply believe that researchers are not doing their statistics well enough.
I'm a layman, so to speak. I'm trained in biology but I have no formal education in statistics or mathematics. I enjoy R and often make an effort to read (and understand...) some of the theoretical foundations of the methods that I apply when doing research. It wouldn't surprise me if the majority of people doing analyses today are actually not formally trained. I've published around 20 original papers, some of which have been accepted by recognized journals and statisticians have frequently been involved in the review-process. My analyses commonly include survival analysis, linear regression, logistic regression, mixed models. Never ever has a reviewer asked about model assumptions, fit or evaluation.
Thus, I never really bothered too much about model assumptions, fit and evaluation. I start with a hypothesis, execute the regression and then present the results. In some instances I made an effort to evaluate these things, but I always ended up with "well it didn't fulfill all assumptions, but I trust the results ("subject matter knowledge") and they are plausible, so it's fine" and when consulting a statistician they always seemed to agree.
Now, I've spoken to other statisticians and non-statisticians (chemists, physicians and biologists) who perform analyses themselves; it seems that people don't really bother too much about all these assumptions and formal evaluations. But here on CV, there is an abundance of people asking about residuals, model fit, ways to evaluate it, eigenvalues, vectors and the list goes on. Let me put it this way, when lme4 warns about large eigenvalues, I really doubt that many of its users care to address that...
Is it worth the extra effort? Is it not likely that the majority of all published results do not respect these assumptions and perhaps have not even assessed them? This is probably a growing issue since databases grow larger every day and there is a notion that the bigger the data, the less important is the assumptions and evaluations.
I could be absolutely wrong, but this is how I have perceived this.
Update: Citation borrowed from StasK (below): http://www.nature.com/news/science-joins-push-to-screen-statistics-in-papers-1.15509