(at around 25 mins) When Ernest Burkhart is presented with a hat, Mollie takes the hat out of the box the right way up. In the next shot, it comes out of the box upside down.
During a bank robbery in the second half of the film, one of the robbers begins walking out of the front door while facing the street. In the next shot, he is facing the other way and walks out of the door backwards.
The radio show references events as late as 1962. But the Lucky Strike package on the set is green; it was switched to white during WWII and never went back. In an interview with Steven Spielberg, director Martin Scorsese addressed the anachronism as deliberate, and is meant to convey that the radio show is not literal, preparing the audience for the fourth wall break.
At the end of William Hale spanking Ernest Burkhart, Robert De Niro hits Leonardo DiCaprio so hard that the paddle breaks, with a splintered crack in the middle of the paddle. An indication that this was accidental comes with De Niro attempting to hide it behind his leg, while the next scene has an unbroken paddle placed on the floor against the podium.
Ernest directs his friend Blackie Thompson to steal his Buick coupe, so he can cash in the insurance. But when Blackie flees the car evading arrest, the Chevrolet bow tie is plainly visible on the grille.
Hale informs Burkhart that he is going to Fort Worth for the stock show. The stock show is held in February, but the trees have leaves on them and nobody is dressed for the usual northern Oklahoma winter.
Near the beginning, as Henry and Ernest arrive home from the train station, they drive through a big herd of cows. In the wide shot, on the left near the road is a brown cow walking. Its shadow develops weird holes where the grass is sunlit where it should remain dark. This reveals the cows to be CGI.
When Mollie's mother dies and meets her ancestors, the bare-chested older male ancestor clearly has a T-shirt tan.
Set in the 1920s, one of the Osage leaders in a tribal meeting says that they are the victims of genocide. The word "genocide" hadn't been coined until 1944 when describing the Holocaust.
During the radio program at the end, one of the characters uses the word "damn," something that would never have been allowed on a broadcast program in America during the time the show appears to be set. (However, director Scorsese has suggested that there are deliberate anachronisms in that scene, and this may be one of them.)
Ernest is told that his uncle "gifted" him something. In that time period, he likely would have been told that his uncle "gave" him something. "Gifted", though in use for about 400 years, has only come into common use since the late 20th century.