Photoluminescence spectroscopy has been employed to study CdSe/ZnSe/ZnMnSe quantum dots. For most of the dots studied here luminescence comes in three spectrally separated features: neutral exciton (X), biexciton (XX), and charged exciton (X_C) states. Spectral properties of X and XX emission are well understood, however, in a marked contrast with previous studies, the observed fine structure of X_C can not be explained within a commonly accepted model of a ground state trion luminescence. We find that at zero magnetic field luminescence from the charged state exhibits fine structure that varies gradually between different dots from a single unpolarized line to a quartet with the maximum splitting of 2 meV. Several models including magnetic polaron formation and double charging have been considered, but a plausible explanation can be given only if one considers the influence of a charge trapped in a nearby dot.