We quantify the linguistic complexity of different languages’ morphological systems. We verify that there is a statistically significant empirical trade-off between paradigm size and irregularity: a language’s inflectional paradigms may be either large in size or highly irregular, but never both. We define a new measure of paradigm irregularity based on the conditional entropy of the surface realization of a paradigm—how hard it is to jointly predict all the word forms in a paradigm from the lemma. We estimate irregularity by training a predictive model. Our measurements are taken on large morphological paradigms from 36 typologically diverse languages.