Deliberate mistake: When Jack and Rose are adrift in the ocean, the stars behind them are not the right ones for that time and date. In fact, it's not any sky ever seen on Earth. The right half of the sky is a mirror image of the left half. The fimmakers deliberately positioned the stars to form the shape of the pendant 'Heart of the Ocean.'
Continuity mistake: When Jack is sneaking over the rail to the first class deck, a boy is preparing to spin his toy top. As Jack leaps the rail and begins walking, the boy throws the top and you see it start to spin. The camera angle then changes to show Jack walking, toward the coat he is about to 'borrow', and in the background you can see the boy again drawing back to toss his toy top. Since you have to meticulously wind the string around the top of the toy, it would be impossible for him to be making a separate toss. It's obviously the same toss filmed from two separate angles.
Continuity mistake: When Jack is playing poker in the beginning of the movie with the Swedish guys and Fabrizio we can see a short shot of his cards. He then takes another card and wins by having a full house. However, there was no way to get a full house with the cards he had by just drawing one more card.
Factual error: Professional Radio Operators hold the key used for Morse code between their thumb and two fingers - they don't tap it, as was shown. Tapping would produce a harsh voice in Morse code.