Wade's collection of military decorations includes the Army's Distinguished Service Medal. That award is reserved only for colonels and generals serving in a position of great responsibility. Wade's role in the film is clearly not that of a former officer of such high rank.
While casually viewing Wade's gun collection, Harlan tells Lonnie that one of the revolvers is a Colt Peacemaker made in 1921. But the gun's year of manufacture would be impossible for Harlan to know without looking up its serial number in an official reference list, which he did not do.
When Harlan and Lonnie are chased away from the campfire, Harlan's mustache is missing in the next scene as they enter the old west town. It is missing from the rest of the film, even though there has been no time for him to shave it off.
October's nail polish changes repeatedly from a dark red color to pale nude, without giving her time to change it. This is particularly noticeable when she is walking in the rain after confronting her father and her nails are red, but then a short while later when she comes out of the rain to talk to Harlan in the diner her nails are pale.
When Harlan takes Lonnie shooting, he uses a gun that takes 6 rounds. After shooting all six out of one and supposedly shooting four from the other, he offers the gun to Lonnie and asks, "Do you want the last two?" However, five shots from the second gun can clearly be heard before he makes the offer.
While Harlan is teaching Lonnie to shoot, he tells the boy that the design of the Colt Peacemaker revolver has never changed since the gun first came out in 1873. In actuality, the Peacemaker underwent several significant changes in the late 1800s. These included a re-designed base pin retainer, an updated ejector rod, and a switch from wood to hard rubber as the standard grip material.
When Lonnie shows Harlan his father's military medals, the second-highest gallantry award of both the Navy (the Navy Cross) and the Army (the Distinguished Service Cross) can be seen. It would be almost impossible for a member of either service branch to win the equivalent medal from the other service.
One of the revolvers in Wade's wall collection is actually a non-firing display model of the Colt Peacemaker. This is evidenced by the horizontal slots in the rear of the cylinder, between the cylinder's flutes. Real Peacemakers do not have such slots.