Networks constructed from credit relationships in the interbank market have been found to exhibit disassortative mixing together with a scale-free degree distribution, in contrast to most social networks that are assortative and not necessarily scale-free. This provokes the question whether generating mechanisms for scale-free networks have enough flexibility to generate both assortative and disassortative structures depending on their parametrization. Using Monte-Carlo simulations, we show that scale-free networks with a small tail exponent tend to be disassortative. However, the simulations indicate also that the level of disassortativity is sensitive to changes in the scaling exponent and the density. A given combination of disassortativity, scaling of the degree distribution, and density in an empirical data set, might be hard or impossible to obtain from any of the known generating mechanisms for scale-free networks.