Marginal Revolution is soliciting nominations for the most famous mistakes of economics. My personal favorite is William Baumol's 1986 American Economic Review article "Productivity Growth, Convergence, and Welfare: What the Long Run Data Show" (link requires JSTOR subscription). In it, he finds that GDP data for 16 countries over the period 1870-1979 validates the convergence prediction of neoclassical growth models. That is, initially poor countries grow faster than rich ones, thereby catching up over time. Looks good, but as Brad DeLong pointed out in "Productivity Growth, Convergence and Welfare: Comment," (AER, 1988 [JSTOR]) the finding suffers from a selection problem. All of the countries in the sample are wealthy in 1979, so, of course, if they started out poor, they caught up. As a way around, with limited data, DeLong instead chooses a sample based on the initial level of income in 1870. Doing so adds several countries like Portugal and Argentina, that did not converge.Some of the nominees at MR have much more cosmic importance, but this case is a good example of how smart people - not just Baumol, but the editors and referees at AER, too - can make mistakes that seem obvious, but only in retrospect after another smart person points them out.
To be fair, it should be mentioned that Baumol did mention the potential issue in a footnote.