Factual error: Throughout the movie, it's been said that President John F. Kennedy's motorcade route had been changed at the last minute. This is not true. The route was set in stone for days. In fact, assassin Lee Harvey Oswald supposedly found out that Kennedy was going to ride right past Oswald's work building from a newspaper days before. He rode the same route the newspaper published.
Character mistake: Jim Garrison and Lou Ivon are discussing the shooting at the book depository and Jim refers to "our friend Clay Shaw" as having introduced General Charles Cavill at the Foreign Policy Commission in New Orleans. However, Garrison does not find out that Clay Bertrand is Clay Shaw until Bill Broussard finds out from his informant and tells Jim in later scenes. (01:18:54)
Answer: Lee Harvey Oswald was very sympathetic to the communist ideal, and lived in Russia for a while. He probably adopted the accent as an affectation towards that end.
Grumpy Scot