We have recently developed a post-processing framework to estimate the abundance of atomic and molecular hydrogen (H i and H 2 , respectively) in galaxies in large-volume cosmological simulations. Here we compare the H i and H 2 content of IllustrisTNG galaxies to observations. We mostly restrict this comparison to z ≈ 0 and consider six observational metrics: the overall abundance of H i and H 2 , their mass functions, gas fractions as a function of stellar mass, the correlation between H 2 and star formation rate, the spatial distribution of gas, and the correlation between gas content and morphology. We find generally good agreement between simulations and observations, particularly for the gas fractions and the H i mass-size relation. The H 2 mass correlates with star formation rate as expected, revealing an almost constant depletion time that evolves up to z = 2 as observed. However, we also discover a number of tensions with varying degrees of significance, including an overestimate of the total neutral gas abundance at z = 0 by about a factor of two and a possible excess of satellites with no or very little neutral gas. These conclusions are robust to the modelling of the H i/H 2 transition. In terms of their neutral gas properties, the IllustrisTNG simulations represent an enormous improvement over the original Illustris run. All data used in this paper are publicly available as part of the IllustrisTNG data release.