The success of the "cluster variation method" (CVM) in reproducing quite accurately the free energies of Monte Carlo (MC) calculations on Ising models is explained in terms of identifying a cancellation of errors: We show that the CVM produces correlation functions that are too close to zero, which leads to an overestimation of the exact energy, E, and at the same time, to an underestimation of - TS, so the free energy F= E - TS is more accurate than either of its parts. This insight explains a problem with "hybrid methods" using MC correlation functions in the CVM entropy expression: They give exact energies E and do not give significantly improved -TS relative to CVM, so they do nor benefit from the above noted cancellation of errors. Additionally, hybrid methods suffer from the difficulty of adequately accounting for both ordered and disordered phases in a consistent way. A different technique, the "entropic Monte Carlo" (EMC), is shown here to provide a means for critically evaluating the CVM entropy. Inspired by EMC results, we find a universal and simple correlation to the CVM entropy which produces individual components of the free energy with MC accuracy, but is computationally much less expensive than either MC thermodynamic integration or EMC.