There is no agreement on the thermal structure of the solar chromosphere. While results of the CO observations and 3D MHD simulations suggest very cool structures in the upper atmosphere, SUMER observations of UV spectral lines is interpreted as signature of a full-time hot chromosphere. We tried to look for cool structures 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 H2v and H2r 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, at a spatial resolution of 1 arcs and a temporal resolution of 5 s, 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 (e.g., FALC), but also with a very cool chromosphere as found in some dynamic simulations.