Identical molecules move with identical velocities when placed in a uniform electric field within a uniform electrolyte. Here we report that homogeneous DNA does not obey this fundamental rule. While most DNA moves with similar velocities, a fraction of DNA moves with velocities that vary within a multiple-fold range. The size of this irregular fraction increases several orders of magnitude when exogenous counterions are added to DNA. The irregular fraction decreases several orders of magnitude when DNA counterions are removed by dialysis against deionized water in the presence of a strong electric field (0.6 kV/cm). Dialysis without the field is ineffective in decreasing the size of irregular fraction. These results suggest that (i) DNA can form very stable complexes with counterions, (ii) these complexes can be dissociated by an electric field, and (iii) the observed non-uniform velocity of DNA is caused by electric-field-induced slow dissociation of these stable complexes. Our findings help to better understand a fundamental property of DNA: its interaction with counterions. In addition, these findings suggest a practical way of making electromigration of DNA more uniform: removal of strongly bound DNA counterions by electro-dialysis against deionized water.