I think they went wrong the moment the Cheka were formed. Once an element of the so-called proletarian revolution put itself above the others is when they reinvented class-distinctions/heirarchy. Communism is the abolition of class. A society with a class structure is a society filled with abuses and corruption, much of it being perpetuated both by the bourgeoisie and by the bureaucrats. The Bolsheviks used violence and fraud in order to elevate themselves, so that they could see the revolution through to success; they must have feared that without their strong guiding hand, the revolution would fall to the reaction. However, just the opposite occurred. Instead the Bolshevik party wrought the destruction of the workers councils in Russia, the subjagation of free thought and therefore workers democracy, the reintroduction of capitalism through the NEP, the reintroduction of a class-society, and finally set the stage for the brutal years under Stalin and the subsequent rule of the Reformists.

But to answer the question in short, the USSR went wrong the moment its leaders thought that a state-based society could somehow give birth to a stateless society.