A police officer tickets Lucy for having expired license plates while her truck is legally parked in the parking lot of the pancake house. Since the parking lot is on private property, and Lucy is not driving the truck when the police officer tickets the expired license plate, he does not have a legal right to write her a ticket. (In some cities police can cite cars on private parking lots.)
Near the end when the patient asks Lucy what he's doing, she tells him he's painting a picture and we see him holding a paint brush. But when we see the picture he "painted" it was actually done in crayon.
At one point in the diner when the waitress takes a guy's order, he says he'll have the pancakes and she takes the menu away from him. In the next shot, he is in the background and still holding the menu open.
In the first shot of Henry, Lucy, Doug and Marlin driving to the Callahan Institute, Henry and Lucy are sitting in the back seat and Lucy's hair is blowing around because the window is down. In all other shots of them driving, Lucy's hair isn't blowing around anymore and no one is ever shown rolling up the window.
As Lucy is driving towards Henry's jumper cable trouble, the camera shows her driving with the driver-side towards the island interior passing some parked trucks. As the shot changes, the close up shot shows the driver-side of the car facing towards the ocean.
After Henry and Lucy burn her journal, it is pouring heavy rain and both of them appear to be dry, but when Henry walks to leave into his truck, Lucy calls after him for one last 1st kiss. She walks out into the rain dry, but in the immediate scene where they finish kissing, they are both completely drenched, they were embracing so there fronts wouldn't be as wet as shown.
After Henry has stitched up Ula's side, it is very red. However, when Ula and Henry are in Henry's that same night or the next, Ula's shark bite looks mostly healed, almost like a scar, without any visible stitches.
When Henry goes to visit Lucy at the Institute, she says, "I don't know who you are, but I dream about you almost every night." Since she has no memories of anything after the accident, it seems that she could not possibly know that she had dreamed about Henry more then once, but she very likely made note of it in her journal.
The wedding portrays Henry as being Jewish, however during the movie it is said that he consumes SPAM, which is mostly pork products and therefore not kosher. However, not all Jews observe the religious laws.
When Ula is on the boat when Henry is repairing it, he has only the slightest white scar. It is only three days after he got bitten and it is only a pale, nearly invisible scar. When he falls through the boards and says his stitches reopened, that is nearly impossible seeing as how it was only a scar and there was no stitches.