<p><em>[eng] Context. Chromospheric fibrils are thin and elongated structures that connect nearby photospheric magnetic field concentrations of</em></p><p><em>opposite polarities.</em></p><p><em>Aims. We assess the possibilities and drawbacks related to the use of current instrumentation and inversion techniques to infer the</em></p><p><em>thermodynamic structure of chromospheric fibrils.</em></p><p><em>Methods.We employed spectroscopic observations obtained in the Ca ii 854.2 nm line with the CRISP instrument at the Swedish 1-m</em></p><p><em>Solar Telescope and in coordination with observations in the ultraviolet Mgii h & k lines taken with the IRIS satellite. We studied the</em></p><p><em>temperature sensitivity of these chromospheric lines to properly invert their spectral profiles with the Stockholm inversion Code and</em></p><p><em>determine the temperature, line-of-sight velocity, and microturbulent velocity of manually traced chromospheric fibrils present in the</em></p><p><em>field of view.</em></p><p><em>Results. Fibril-like structures show a very particular dependence of their temperature as a function of the position along their length.</em></p><p><em>Their temperatures at the detected footpoints are, on average, 300K higher than the temperature at the midpoint. The temperature</em></p><p><em>variation appears to be almost symmetrical in shape, with partially traced fibrils showing a similar trend for the temperature variation.</em></p><p><em>Additionally, the response of the Ca ii 854.2 nm line core to variations of the temperature for the inverted models of the atmosphere</em></p><p><em>in fibril areas seems to be insu cient to properly resolve the aforementioned temperature structure. Only the addition of more temperature</em></p><p><em>sensitive lines such as the Mgii h & k lines would make it possible to properly infer the thermodynamic properties of</em></p><p><em>chromospheric fibrils. Comparisons between the results obtained here and in previous studies focused on bright Ca ii K fibrils yield</em></p><p><em>great similarities between these structures in terms of their temperature.</em></p>