Andrew Gelman discusses research on attitudes toward gay marriage, by state, and notes this graph in particular, which shows the change in opinion over the last 15 years:
Critically, he points out that the states which experienced the greatest change in attitude were the ones that already were most receptive. A naive analysis of the data would have missed this effect, which is an important one (Andrew proposes a few testable hypotheses for it). Once a statistician knew the effect existed, however, it would be relatively simple to incorporate it into a model.
The effect stresses the importance of properly exploring data (visually, not necessarily quantitatively) rather than simply chucking it into a regression. Even a simple analysis like this one can reveal features that would take a specially chosen model to identify otherwise.