In the miniature people scene, after Larry breaks free from his upper body restraints, the roping around his ankles is still in place. In the shot just before the Romans attack him with fire balls, the ropes are shown around his ankles, still in place. However, after the Romans begin to attack him, Larry is able to get up and run away with ease- there is no shown attempt to break the restraints around his ankles.
When Larry and Rebecca were talking outside, you can clearly see a mike drop down in the shot, bob up and down and then taken out of the shot. At the end with career day at school the mike appears again.
The day after Larry's first night, Rebecca tells a children's tour that the lion is the "king of the jungle". The lion lives in plains and open grassland, and is unsuited for either hunting or concealing itself in jungle or forest. It's true appellation is "king of beasts".
When Larry is trying to convince Rebecca that the exhibits at the museum come to life at night, they stop to talk directly in front of the Neanderthal exhibit, but when Rebecca walks away, they are next to Theodore Roosevelt.
In the scene where the Mayans attack Larry by shooting poison darts at him, the darts leave two very large welts on his face that also appears to have caused him to lose feeling there as well. But as the camera scene cuts to him trying to defend himself from a second, larger attack the welts are gone from his face and he takes about a dozen darts in his hand, this time with no apparent affect.
When Larry and Nick are in Central Park at the beginning of the movie, the building with two towers is visible behind Larry. When they turn to leave the park, the building is still behind Larry. However, in the immediate scene following, the same building is now in front of Nick and Larry.
Theodore Roosevelt was known to have a high-pitched, squeaky voice due to being a severe asthmatic. However, as was later established, he was only a wax figurine of Roosevelt, made to appear very heroic, so it's not unreasonable that he'd have a deeper, more resonant voice, not unlike Robin Williams.
The most famous Jedediah Smith was a fur trader, hunter and explorer, not a cowboy and railroad-builder--he died just as railroads were being invented. But the name was probably common enough in the 19th century that many men were named Jedidiah Smith. The movie's "Jedidiah" is a representative period figure, not a depiction of a historic person. It's also possible that the toy-come-to-life heard the name and liked it, and so chose it for himself.
Anything relating to what the magically-enlivened museum pieces can or cannot do. Since they are fantasy beings, they can do whatever the writers say they can do, even when the logic appears contradictory.
In the closing sequence when the museum characters are dancing and playing soccer, the cavemen are still marveling at fire, making noises, you can obviously see one of them has a filling - something neither a wax figure or a genuine caveman would have.