Observing non-Gaussianities in the timing residuals of Pulsar Timing Arrays (PTAs) has recently attracted attention as a potential discriminator between astrophysical and cosmological origins of the observed Gravitational Wave (GW) signal. In this work, we show that even in an idealized signal-dominated setup, after decorrelating the data to avoid spurious detections, statistical tests applied to PTA data cannot distinguish between a Gaussian and a non-Gaussian amplitude distribution of the GWB in a model-agnostic way. In particular, without making strong assumptions on the GW spectrum or the properties of the population, the sensitivity to any distinctive non-Gaussian feature is washed out.