Double-stranded DNA is among the stiffest biopolymers, whose bending propensity crucially influences many vital biological processes. It is not fully understood which among the two most likely forces, electrostatic self-repulsion or the compressive base pair stacking, plays a dominant role in determining the DNA's unique rigidity. Different theoretical and experimental studies led so far to contradictory results on this issue. In this Communication, we address this important question by means of Molecular Dynamics (MD) simulations using both atomistic and coarse-grained force fields. Using two independent sets of calculations, we found that electrostatic and nonelectrostatic effects play a comparable role in maintaining DNA's stiffness. Our findings substantially differ from predictions of existing theories for DNA rigidity and may indicate that a new conceptual understanding needs to be developed.