We investigate the efficiency of screening mechanisms in the hybrid metric-Palatini gravity. The value of the field is computed around spherical bodies embedded in a background of constant density. We find a thin shell condition for the field depending on the background field value. In order to quantify how the thin shell effect is relevant, we analyze how it behaves in the neighborhood of different astrophysical objects (planets, moons or stars). We find that the condition is very well satisfied except only for some peculiar objects. Furthermore we establish bounds on the model using data from solar system experiments such as the spectral deviation measured by the Cassini mission and the stability of the Earth-Moon system, which gives the best constraint to date on $f(R)$ theories. These bounds contribute to fix the range of viable hybrid gravity models.
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