The chain increment method and configurational bias Monte Carlo methods are used to test the approximations made in the derivation of the generalized Flory-Dimer (GF-D) theory for tangent hard sphere chains. Insertion probabilities and residual chemical potentials are calculated for hard chain fluids containing chains of length n=4, 8, 16, and 32 at monomer densities, rho(M), up to 0.8. We find that the largest errors in the GF-D theory are those associated with assuming that the probability of inserting a monomer into a chain fluid is approximately equal to the probability of inserting a monomer into a monomer fluid, as predicted by the Carnahan-Starling equation of state. The errors in the incremental compressibility factors of the second segment associated with assuming that the conditional probability of inserting a second bead next to the first bead in a chain fluid is approximately equal to the probability of inserting a second bead next to the first bead in a dimer fluid as predicted by combining the Carnahan-Starling and Tildesley-Streett equations of state are relatively small. Consistent with the findings of Mooij and Frenkel, we find that these two approximations lead to an overprediction of the incremental contributions to the compressibility factor. Despite the overprediction of the incremental contributions to the compressibility factor of the first segment, the GF-D equation of state accurately predicts the compressibility of hard chains; this accuracy is traced to (1) the insensitivity of the compressibility factors to errors in the insertion probability adn (2) cancellation of errors in the incremental compressibility factor of the first segment with small cumulative errors in the incremental compressibility factors of the third and subsequent segments.