We study chromospheric emission to understand the temperature stratification in the solar chromosphere. We observed the intensity profile of the Ca II H line in a quiet Sun region close to the disk center at the German Vacuum Tower Telescope. We analyze over $10^5$ line profiles from inter-network regions. For comparison with the observed profiles, we synthesize spectra for a variety of model atmospheres with a non local thermodynamic equilibrium~(NLTE) radiative transfer code. A fraction of about 25% of the observed Ca II H line profiles do not show a measurable emission peak in $H_{2v}$ and $H_{2r}$ wavelength bands (reversal-free). All of the chosen model atmospheres with a temperature rise fail to reproduce such profiles. On the other hand, the synthetic calcium profile of a model atmosphere that has a monotonic decline of the temperature with height shows a reversal-free profile that has much lower intensities than any observed line profile. The observed reversal-free profiles indicate the existence of cool patches in the interior of chromospheric network cells, at least for short time intervals. Our finding is not only in conflict with a full-time hot chromosphere, but also
with a very cool chromosphere as found in some dynamic simulations.