As the Soviet Union became a force in world chess after World War II, numerous westerners accused the Russians of collusion. With so many Soviet grandmasters participating in high-level tournaments, it was easy to stage results to ensure victory for a specific player.
Here's the thing: as likely as the Soviets were at colluding against the west, they were as likely to collude against each other. Take David Bronstein, for one.
Bronstein is considered one of the greatest chess players of all time, but he never won the world championship. In fact, he only qualified for it once, tying Mikhail Botvinnik in the final. By the rules, since Botvinnik was the reigning champion, he kept the crown. And Bronstein, despite all his prowess, never got close to challenging again.
To qualify for the championship match, Bronstein had to pass the Soviet-filled Candidates Tournament. So why did they collude against him? There are many theories. One is that they preferred someone from the Russian Republic, and not the Ukrainian-born Bronstein. Another is that his personality was too outgoing and not "Soviet enough". Another is that he was related to Leon Trotsky (real last name: Bronstein).
Of course, being Jewish didn't exactly help...