These results are interesting because they demonstrate that, when dealing with the optimization of approximated models, there will be inevitably intrinsic limits due to degeneration of the optimal set of parameters that satisfy the conditions that are posed. In such a case, further improvements cannot be achieved without introducing additional degrees of freedom that can decouple in some way such composite variables into the fundamental ones allowing to fine-tune the model. One way could be, e.g., to add some higher (quantum) level additional constraint that allows to decouple the dependence on the charge (q) from that of geometry (d) in the solution. However, the geometry and electrostatics of QM water molecules are so different from those of these AA models, that tests in this sense proved inefficient. In the case of the classical water models studied herein, reaching higher precision thus requires expanding the model's representation by adding additional "classical" degrees of freedom, for example, allowing for a more flexible and accurate description of the system. This is exactly the case of the higher precision that can be achieved by, e.g., 4-or 5-site models. Moreover, altering the degrees of freedom can have important effects for example on subtle dynamical mechanisms associated with water reorientational dynamics as recently shown. However, these results are also interesting for the development of approximated molecular models in general. Recently, we have observed similar intrinsic limitations also in the optimization of, e.g., coarse-grained models of a variety of other molecular systems. This observation serves as a valuable lesson for developing models of all kinds, not just in the context of water simulations. Such inherent limitations and these challenges encountered in optimizing approximated models demonstrate the importance of considering the complexity of the system being studied, and the type of information lost with approximated molecular models. The integration of multiple references, and in particular combining bottom-up and top-down microscopic/macroscopic-level information in the training-set can improve the efficiency and robustness in the models' optimization. Nonetheless, the results discussed herein also offer an unambiguous example of how, understanding the physical limits of approximated models, can provide a precious knowledge for guiding future research towards more robust and reliable modeling approaches.