We analyzed the ability of variants of the SPC/E and TIP4P types of water models to describe the temperature dependence of their second virial coefficients, liquid-vapor phase envelopes, and corresponding coexistence vapor pressure. We complete the characterization of the two most promising models by testing their adequacy to predict the structure of the 13 known crystalline phases of ice by (Parrinello-Rahman) isothermal-isobaric Monte Carlo simulations. While these models perform well for the description of properties to which their force fields were fitted (density, heat of vaporization, structure at the level of pair correlations), their transferability to the entire phase diagram is unsatisfactory, i.e., none could significantly mitigate the shortcomings of the original models. In fact, the most appropriate alternative appears to be the TIP4P-EW model, i.e., the recent reparametrization of the original TIP4P water model. Model parametrizations aimed at improving the description of ice behavior fail even in the description of the liquid phase. (c) 2006 American Institute of Physics.