Models of biochemical regulation in prokaryotes and eukaryotes, typically consisting of a set of first-order nonlinear ordinary differential equations, have become increasingly popular of late. These systems have large numbers of poorly known parameters, simplified dynamics, and uncertain connectivity: three key features of a class of problems we call sloppy models, which are shared by many other high-dimensional multiparameter nonlinear models. We use a statistical ensemble method to study the behavior of these models, in order to extract as much useful predictive information as possible from a sloppy model, given the available data used to constrain it. We discuss numerical challenges that emerge in using the ensemble method for a large system. We characterize features of sloppy model parameter fluctuations by various spectral decompositions and find indeed that five parameters can be used to fit an elephant. We also find that model entropy is as important to the problem of model choice as model energy is to parameter choice.