tl;dr Sometimes a mathematical result is strikingly contrary to generally held belief even though an obviously valid proof is given. Charles Stein of Stanford University discovered such a paradox in statistics in 1995. His result undermined a century and a half of work on estimation theory. (Efron & Morris, 1977, p. 119)
The James-Stein estimator leads to better predictions than simple means. Though I don’t recommend you actually use the James-Stein estimator in applied research, understanding why it works might help clarify why it’s time social scientists consider defaulting to multilevel models for their work-a-day projects.