The German Democratic Republic, People's Republic of China even the USSR all have democratic names and even sensibilities. They never say anything bad about democracy but instead attach words like imperialist, fascist or revanchist to countries and ideas that trouble them. Is their concept of democracy different than ours? Was there once suffrage among workers and revolutionaries like in republican Spain? Or was it always a meaningless title?
The basic concept behind socialism and communism is radical democracy, that not only should workers have a vote in how their nation is run, but democratic control over the private sector as well.
Consider the USSR. "Soviet" in Russian was originally just a word for "council," in the sense of a town or community council. The labor movement in Russian gave rise to various democratically elected councils of workers in various towns and cities, and when the Czar's authority started to break down under the weight of WWI many people turned to these councils for guidance. The entire history of the Russian revolution is a little complex to get into here, but yes, most of the movements that produced "communist" countries had their roots in democratic processes of one form or another.
What tends to happen to these countries, though, is that stresses both external or internal tend to derail the democratic process. This can take a number of forms. The Bolsheviks in Russia became increasingly authoritarian as a result of these stresses, but other attempts at socialist nations simply buckled under counterrevolutionary forces, as Allende's Chile did to Pinochet.