We demonstrate that pairwise contact potentials alone cannot be used to predict the native fold of a protein. Ideally, one would hope that a universal energy function exists, for which the native folds of all proteins are the respective ground states. Here we pose a much more restricted question: Is it possible to find a set of contact parameters for which the energy of the native contact map of a single protein (crambin) is lower than that of all possible physically realizable decoy maps? The set of maps we used was derived by energy minimization (not by threading). We seek such a set of parameters by perceptron learning, a procedure which is guaranteed to find such a set if it exists. We found that it is impossible to fine-tune contact parameters that will assign all alternative conformations higher energy than that of the native map. This finding proves that there is no pairwise contact potential that can be used to fold any given protein. Inclusion of additional energy terms, such as hydrophobic (solvation), hydrogen bond, or multibody interactions may help to attain foldability within specific structural families.