For molecules containing second row atoms, unacceptable errors have been found in extrapolating dissociation energies calculated with the standard correlation consistent basis sets to the complete basis set limit. By carefully comparing the convergence behavior of D-e(O-2) and D-e(SO), we show that the cause of these errors is a result of two inter-related problems: near duplication of the exponents in two of the d sets and a lack of high-exponent functions in the early members of the sets. Similar problems exist for the f sets (and probably in higher angular momentum sets), but have only a minor effect on the calculated dissociation energies. A number of approaches to address the problems in the d sets were investigated. Well behaved convergence was obtained by augmenting the (1d) and (2d) sets with a high-exponent function and by replacing the (3d) set by the (4d) set and the (4d) set by the (5d) set and so on. To ensure satisfactory coverage of both the L and M shell regions, the exponents of the new d sets were re-optimized. Benchmark calculations on Si-2, PN, SO, and AlCl with the new cc-pV(n+d)Z sets show greatly improved convergence behavior not only for D-e but for many other properties as well.