Continuity mistake: When Emily is spying on Cody on the sightseeing bus, she sits down and the camera goes to a close up of her face and we see small worm like camera device comes out of the right side of her sunglasses, but when it cuts to the next shot, further back from her face, we can see the device has disappeared even though no time has passed between shots.
Factual error: When Derek is calculating how much explosive Mentos to put in Cody's mouth to get the microchip out, he says to himself, "a thousand nanograms equals 1 microgram". While this measurement is correct, it is highly unlikely that any explosive would be of use in even milligram quantities. So 1/1000 of a milligram (a microgram) would be virtually useless, and measurements of 1/1,000,000 of a milligram (a nanogram) would be even less than useless. He couldn't have been calculating explosives in such small quantities. As well, Mentos mints are about 37.5 g (http://home.socal.rr.com/mentosfaq/candies.html), and he then cuts it into approximately 1/8 - micrograms would measure 1/250 of a mint, and nanograms would be 1/2500 of a mint.
Plot hole: The secret CIA base at Kamp Woody was underground. Kids searching for years have combed the camp, but never found it, but when Cody is going down the escalator, you can see bright skylights on the roof. If the sun is able to shine through, that means the skylights aren't hidden, and the kids would be able to find the camp.
Revealing mistake: When the musicians stop playing and the world leaders start getting up from the table, it goes to a shot spanning the entire table, but also showing the musician's stage. This is where you see the mistake, as you see Berkhamp suddenly at the front of the stage wearing his headband and dancing, when it cuts away from that shot, it shows Berkhamp is still sitting behind his instrument with his headband still off. Then we find that out of place shot was recycled from when Berkhamp actually was onstage and dancing as we see the exact same shot again less than a minute later, this time the shot is where it's supposed to be however this is still a mistake and quite an odd one.
Continuity mistake: In the shot where the dog is about to play the piano, before the dog starts playing he pours some whiskey into a glass. When the dog plays the piano however the glass has gone from the table. But then when the dog has finished playing the piano, the glass is on the table again.
Continuity mistake: When Cody, Derek and Kumar are on the observation wheel in London, Derek gives Cody a newspaper. You see through the glass that they're a few yards above ground level as if they've just got on the wheel. Then they cut to a shot of the newspaper for a couple of seconds while Cody reads a headline, then it cuts back to a shot where we can see the outside and suddenly they're high above the city. Dialogue indicates that no time has passed between shots so there's no possible way that the observation wheel could have lifted them that high in such a small amount of time, especially because in both shots, it's clear the observation wheel is not moving.
Plot hole: When Cody first discovers what the explosive Mentos mints do, the explosions are just minor. Later on in the movie, the Mentos are treated as though they create big explosions. At the end of the movie, Cody throws a Mentos, which his brother licked on, out of the car and into a lake; the explosion was bigger than the explosion made when they were first used.