As Eleanor and Joe are talking by the window in the airport lounge, the people sitting behind them change with almost every cut.
Private Bailey's name tag and service tag switches from left side to right side repeatedly at the dinner table.
The narration is inconsistent. Sometimes it is being narrated by Rags; other times the narrator refers to Rags in the third person (or dog, as the case maybe).
The puppy shown at the beginning is definitely a Saint Bernard puppy, while the adult dog it should have grown into is an unknown kind, but obviously different. Even the muzzle colors are different.
At around 01:20:00 into the movie the amount and speed of the snowing outside is different at several windows.