When Alex is walking through the "spy lab" he and his guide walk past a man putting a manikin head on a body, in the next cut when Alex and girl are on the stairs the head is off and the man is picking it off the floor as if it had blown off.
When Alex and Kelly are in the sewer they are wearing their bow ties. While they are opening up to each other their ties are off, but when they emerge from the sewer the ties are back on. Taking them off makes sense, but why put them back on again and smarten themselves up *before* climbing out?
When Kelly gets a call from the president to get his assignment, the girl sitting next to him on the left is wearing a black dress. In the next shot, a few seconds later, another girl wearing blue dress sitting next to Kelly - she was sitting behind him and the black dress girl is gone.
When Kelly and Alex leave the warehouse to go to the party at 20:00 they enter a Mercedes Benz S-Class Limousine (W140). However, as they approach the entrance to the party, when the scene is inside the car, the dashboard design is that of a Mercedes-Benz E-Class (W124). Also, the car that Alex and Kelly get out of is not a super-stretch Mercedes limousine with darkly tinted windows. It appears to be a standard stretch sedan likely a Mercedes-Benz E-Class (W124).
When Kelly and Alex are sitting in the BMW Convertible on the car-transporter they are driving to the north on the quay of the Pest side of the river, but when they drive off the car-transporter they are heading to the south at the Buda side. Note the "Freedom statue" on the right at the top of the hill (Gellérthegy) and the bridge (Erzsébet-híd) in front of him.
After Alex puts the pilot in Kelly's plane to sleep, he descends the plane to persuade Kelly to go to the party at 8pm, however, a steep descent as shown in the film is not only impossible, as they went from apparently 36,000ft to around 1, 000ft as shown on the altimeter a few seconds later, but would also make the plane exceed its structural load limit and disintegrate. In addition, it would make everything inside the plane weightless due to the negative G-forces.
Near the end of the movie, Alex sinks the plane into the river, and he and Kelly swim up to the surface. However, when the camera pulls out at the end of the scene, we can clearly see that the two guys are not floating but just standing in the chest-high water. The plane, which was supposed to be sunk, is still clearly visible behind them.
In the opening of the scene at the top of the bridge in Budapest, the water in the bottom-left corner of the screen has ripples that reflect the lights of the city. However, due to the fact that this is a soundstage backdrop, the ripples do not move.