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It's not a Buick. The grille, which is not unlike that of a Triumph Dolomite, is rather distinctive, and the Triumph grille was a copy of the one used by Hudson.
I think you'll find that your mystery car is a 1936 or 1937 Hudson Terraplane.
http://en.wikipedia.org/wiki/File:TerraplaneConvertible-RockvilleMDshow2007.JPG