Ths sound of the blood dripping from Max's arm is heard when he's on the computer. When the dog just starts licking the blood on the carpet you still hear blood dripping, but you cannot see it falling and it definitely won't sound like its falling in a puddle if the dog stands over the only puddle in the room.
Max's character is bleeding profusely while he is typing on the computer, so much so it saturated his shirt sleeve and the dog. and pooled on the carpet. When he decides to leave hurriedly and subsequently loses deliberately at Jenga, there is no sign of blood anywhere. In later scenes there in also no blood on his clothes and he seems to have full use of the arm in later fight scenes.
After Annie, Max and Brooks run away from the gangsters in the bar one of them is almost hit by a car. In the background you can see Annie open the car door. After the cut, she opens the door again to get into the car.
When Annie and Max are talking in the bar, the camera switches back and forth between them several times. From one side Annie's sweater sleeve extends beyond her jacket, from the other view it doesn't.
When they find his brother n the back room of the bar, Rachel McAdams shoots the .38 revolver into the ceiling and it shatters a light fixture.. That scares her and she drops the handgun onto the floor. The revolver then shoots another cartridge and he bullet goes into Max's arm. Problem is, revolvers do not have the same abilities of semi-automatic weapon without rotating the cylinder and the hammer cocking and releasing via the trigger pull.
When we first see Gary in uniform, the US Flag patch on his right shoulder shows the blue field on the left. This is the proper display of the US Flag EXCEPT on uniform patches. The blue field is always facing forward on right sleeve patches.
In the end credits scene which shows Gary's game plan, VHS cover boxes of The Green Mile and Braveheart are shown which seem to only fit one videocassette each. However, both movies were long enough to require two tapes and thus the cover boxes would be thicker.
Max is shot and stabbed in the forearm early in the film, yet only three months later his wounds have completely healed with no bandages or scars. There's no chance wounds this severe would heal that quickly.
When Max logs into Gary's computer, the password is visible as he is typing it. Any modern computer concerned with security -- of which a police computer would certainly be one -- does not display the password as you type.