Competitor thinks that we Dos-Attacked them, but we didn’t

Guys from competitor startup wrote to us “Please stop Dos Attacking our website. Our website does not work for 5 days. If you won’t stop we will start to Dos you as well. We don’t have any other competitor then you, that is why we are sure that you are doing it.” After couple hours of chat we explained them that we did not attack their website (that is true).Then we started to talk about Dos attacks and how we can protect our website. Our competitor said that professional webhosting would protect website from Dos Attacks. Is that true that pro hosting can protect us from Dos Attack?