Negative differential capacitance (NC) occurring at low frequencies in organic light-emitting diodes (OLEDs) is a poorly understood phenomenon. We study the origin of the NC effect by systematically varying the number of electron traps in OLEDs based on the polymeric semiconductor poly(p-phenylene vinylene). Increasing the electron trap density enhances the NC effect. The magnitude and observed decrease of the relaxation time is consistent with the (inverse) rate of trap-assisted recombination. The absence of NC in a nearly trap-free light-emitting diode unambiguously shows that trap-assisted recombination is the responsible mechanism for the negative contribution to the capacitance in bipolar organic diodes. Our results reveal that the NC effect can be exploited to quantitatively determine the number of traps in organic semiconductors in a nondestructive fashion.