Extremely narrow election outcomes—such as could be reversed by rejecting a few thousand ballots—are likely to trigger dispute over the results. Narrow vote tallies may generate recounts and litigation; they may be resolved by courts or elections administrators (e.g., Secretaries of State disqualifying ballots) rather than by voters; and they may reduce the peacefulness, perceived legitimacy, or predictability of the transfer of political power. In this paper we evaluate the probability of such disputable US presidential elections under a hypothetical National Popular Vote versus the current Electoral College system. Starting from probabilistic simulations of likely presidential election outcomes that are similar to the output from election forecasting models, we calculate the likelihood of disputable, narrow outcomes under the Electoral College. The probability that the Electoral College is decided by 20,000 ballots or fewer in a single, pivotal state is greater than 1-in-10. Although it is possible in principle for either system to generate more risk of a disputable election outcome, in practice the Electoral College today is about 40 times as likely as a National Popular Vote to generate scenarios in which a small number of ballots in a pivotal voting unit determines the Presidency. This disputed-election risk is asymmetric across political parties. It is about twice as likely that a Democrat's (rather than Republican's) Electoral College victory in a close election could be overturned by a judicial decision affecting less than 1,000, 5,000, or 10,000 ballots in a single, pivotal state.