Communism is in theory democratic, as it entails the control of social and economic affairs by the working class as a whole and also the abolition of class hierarchy, and therefore rule by the people. In practice however, self-labeled Communist states have failed to be truly democratic in virtually every case despite their claims to be (notice that virtually every Communist country has "republic" or "democratic republic" somewhere in its official name).

To help explain why this is the case, I'm going to focus on the USSR because 1) it's the country I know the most about and 2) it was the model for all later Communist governments that would form during the twentieth century. (This response will also assume here that "Communism" refers to Marxism-Leninism and not any of the various other less prominent forms of communism such as anarchist communism.)

The first thing that is important to note is that the USSR did have elections for the offices of political leaders, and these leaders probably did genuinely believe for the most part that the political system they existed in was a democratic one. However, elections in the USSR worked in a nested fashion whereby ordinary citizens would elect the members of their local council (or soviet), then those elected members of all the local councils within a given district would vote for the members of their district council, then the members of all the district councils within a given region or territory would elect the members of their regional/territorial council, then they would elect the members of their national council, and the members of all the national councils would elect the members of the union council (Politburo).

This was in theory supposed to carry popular opinion up through the political hierarchy to the highest levels while filtering it through the responsible caution of elected representatives, but due to the number of layers of nested representation, public opinion had only a marginal effect on actual government policy. Because all officials above the local level competed for votes not among the general public but among their fellow council members, each additional layer of representation meant a greater degree of estrangement from the popular will.

This issue was compounded by the fact that authority within the political hierarchy emanated from the top down (i.e. national soviet officials had authority over regional/territorial, district, & local officials, Politburo officials had authority over everyone below them), meaning that the amount of power you had within the system was directly proportional to how insulated you were from popular opinion. What's more, during the Russian Civil War all other parties besides the Bolshevik (Communist) party had been denounced by Lenin and other leading Bolshevik officials as counterrevolutionaries and traitors and were banned, and membership in the Communist Party was made a requirement for participation in the government from then on. Since higher-level councils reserved (and liberally exercised) the right to expel lower-ranking members from the Communist Party if they saw fit, this effectively meant that political officials were able to pick and choose their own voters based on their own standards of eligibility, further undermining the power of the vote. The atmosphere of war also enabled the government to enact harsh and severe restrictions on freedoms of speech and the press, thereby further reducing what limited control ordinary citizens had in influencing public policy.

Due to these factors, politics in the USSR by the time of Lenin's death were shaped largely by the competition of cliques of political elites within the Politburo, and not by public opinion. Stalin rose to the position of General Secretary of the Communist Party largely due to his reputation as a moderate between the Left-wing clique (led by Trotsky) and the Right-wing clique (led by Bukharin), and he thereupon consolidated power under his own platform by leading various purges against oppositionists on both sides by painting them as traitors. The near-annihilation of the Soviet Union by the White Army during the Russian Civil War had produced an atmosphere of fear, paranoia, and distrust within Soviet political culture, leading to an increased emphasis on unity and loyalty to the Party and its leadership (the groundwork for which had already been laid out by Lenin's theory of "Democratic Centralism"), which would be the driving concern of political elites during the Stalinist period.

J. Arch Getty & Oleg V. Naumov, The Road to Terror: Stalin and the Self-Destruction of the Bolsheviks, 1932-1939 (New Haven and London: Yale University Press, 1999)

That's the basic rundown, other Marxist-Leninist regimes have had similar contradictions of democracy due to their adoption of Lenin and Stalin's theories of government, but I'm not as sure on the specifics as I know there was a great deal of variation from country to country. As for the second part of your question (Did any truly democratic Communist countries exist?), it all of course depends on how you define "democratic" and how you define "communist," but within the realm of Marxist-Leninist political parties I think a few good cases to look into would be the government of Salvador Allende in Chile, who came to power in a democratic election (but was overthrown three years later in a CIA-backed military coup), and the Sandinista National Liberation Front in Nicaragua, which held some fairly democratic parliamentary elections in 1984 and 1990 after overthrowing the Somoza dynasty in 1979, and which returned to power democratically in 2011 (after losing the 1990 election to UNO) and remains the current ruling party in Nicaragua.