We present experimental data on the time and radius-dependent length distributions of Au-catalyzed InAs nanowires grown by metal organic vapor phase epitaxy. We show that these distributions are not as sharp as commonly believed. Rather, they appear to be much broader than Poissonian from the very beginning and spread quickly as the nanowires grow. We develop a model that attributes the observed broadening to the diffusion-induced character of growth. In the initial growth stage, the nanowires are fed from their entire length, leading to a Polya-like length distribution whose standard deviation is proportional to the mean length. After the nanowire length exceeds the adatom diffusion length, the growth acquires a Poissonian character in which the standard deviation scales as a square root of the mean length. We explain why wider nanowires have smaller length dispersion and speculate on the length distributions in Au-catalyzed versus self-catalyzed growth methods.