The magnetic structures of the Sun are very inhomogeneous, with irregularities smaller than the smallest sizes that we resolve from Earth. Such irregularities are not properly accounted for by standard magnetic field diagnostic techniques. We have identified a quantitatively important bias that has remained unnoticed hitherto. Intense magnetic fields embedded in inhomogeneous magnetic structures produce little light and easily escape detection. These elusive magnetic fields, which cheat standard observing techniques, seem to be common. We estimate that they carry at least half of the solar magnetic flux. Should the bias be so severe, it would cast doubts on the present interpretation of many solar magnetic phenomena. Since magnetic field measurements in solar-type stars reproduce solar methods, they are liable to the same systematic errors.