When Barret is buying the newspaper, in the long shot the newspaper is the Los Angeles Chronicle and the banner headline is "Space Conference Opens". But when he holds up the paper to examine the date, the newspaper is named The Star Dispatch and the banner headline is "New Bonds Voted by Citizens".
When Jim is looking at the Cinnamon's photo he removes his cigarette and holds it in his left hand, the next shot is the photo of Willy, but there is no cigarette in Jim's hand.
They state that the statute of limitations is a particular time of day. Actually, the statute of limitations is based only on the day and doesn't run out until midnight of the specified day.
When Barney is creating the fake human-freezing machine, he begins to test out its refrigeration capabilities. The camera moves to a gauge that reads "Chamber Temperature Centigrade" and it is reading at approximately 51 degrees, already rapidly moving down. The problem is that 51 degrees centigrade is 124 degrees Fahrenheit, and with this fake machine it would have started roughly at room temperature.
When Barret is following Jim to the staged blackmail payoff, Jim watches him in the rear view mirror. But the image is flipped so Barret appears to be in a right-hand drive vehicle, which is not how such a mirror image would look.
To convince Barret that the statute of limitations had run out, they arrange for him to see a newspaper dated August 18, 1968 - "one day over" according to Phelps. But earlier in the story Barret had seen a faked glossy magazine dated December 1968. (Monthly magazines often post-date their cover dates to maintain shelf life, but this seems excessive).