Comment by gwern

That's a good point. Another one I forgot to make: given the established empirical reality of 'everything is correlated', if you find a variable which does in fact seem to be independent of most or everything else, that alone makes that variable suspicious - it suggests that it may be a pseudo-variable, composed largely or entirely of measurement error/randomness, or perhaps afflicted by a severe selection bias or other problem (such as range restriction or Berkson's paradox eliminating the real correlation).

Somewhat similarly, because 'everything is heritable', if you run into a human trait which is not heritable at all and is precisely estimated at h^2~0, that cast considerable doubt on whether you have a real trait at all. (I've seen this happen to a few latent variables extracted by factor analysis: they have near-zero heritability in a twin study and on further investigation, turn out to have been just sampling error or bad factor analysis in the first place, and don't replicate or predict anything or satisfy any of the criteria you might use to decide if a trait is 'real'.)