Whenever I teach growth theory, I like to compare the Canadian experience with that of Argentina. Up until the 1930's, the two countries followed very similar paths: foreign investment financing the development of resource-based economies. But then the 1930's happened, and Canada and Argentina parted ways.
This is the best graphical demonstration that starting points are not destiny (the data are from Angus Maddison):
During the 65 years between 1870 and 1935, Argentina kept pace with Canada. Since then, Argentina's income per capita increased by a factor of 3, less than half that of Canada.
It's hard to see how that gap could be explained by anything other than the unhappy choices made by Argentina's political classes over the past four generations. Which makes me think that the answer to Dani Rodrick's question is a despairing 'yes'.
Update: Brad DeLong has just reposted his 1991 piece with Barry Eichengreen on the decisions Argentina took in the 1930s and afterwards.