We obtain a necessary and sufficient condition under which random-coefficient discrete choice models, such as mixed-logit models, are rich enough to approximate any nonparametric random utility models arbitrarily well across choice sets. The condition turns out to be the affine-independence of the set of characteristic vectors. When the condition fails, resulting in some random utility models that cannot be closely approximated, we identify preferences and substitution patterns that are challenging to approximate accurately. We also propose algorithms to quantify the magnitude of approximation errors.