[t_mann]

> However, both elevated CRP and higher fasting insulin are correlated with obesity, so controlling for these variables seems misleading.

Well: https://en.wikipedia.org/wiki/Multicollinearity

Summary: highly correlated variables don't fundamentally mess up the inference (coefficient estimates are still unbiased), the estimation becomes numerically unstable, however, and individual coefficients might be messed up because of that. This is reflected in the standard errors, so you're more likely to miss effects (type II error).