Recent experiments suggest that the photoluminescence line width of CdSe and CdSe/CdS nanoplatelets (NPLs) may be broadened by the presence of shakeup (SU) lines from negatively charged trions. We carry out a theoretical analysis, based on effective mass and configuration interaction (CI) simulations, to identify the physical conditions that enable such processes. We confirm that trions in colloidal NPLs are susceptible of presenting SU lines up to one order of magnitude stronger than in epitaxial quantum wells, stimulated by dielectric confinement. For these processes to take place trions must be weakly bound to off-centered impurities, which relax symmetry selection rules. Charges on the lateral sidewalls are particularly efficient to this end. We propose that the broad line width reported for core/shell CdSe/CdS NPLs may relate not only to SU processes but also to a metastable spin triplet trion state. Understanding the origin of SU processes opens paths to rational design of NPLs with narrower line width.