Fibrils are thin elongated features visible in the solar chromosphere in and around magnetized regions. Because of their visual appearance they have been traditionally considered a tracer of the magnetic field lines. In this work we challenge that notion for the first time by comparing their orientation to that of the magnetic field, obtained via high-resolution spectro-polarimetric observations of Ca II lines. The short answer to the question posed in the title is that mostly yes, but not always.