Ben Kenobi: Obi-Wan Kenobi? Obi-Wan. Now there's a name I've not heard in a long, long time. A long time.
Luke Skywalker: You do know him then?
Ben Kenobi: Of course I know him...he's me. I haven't gone by the name Obi-Wan since, oh, since before you were born.
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Deliberate "mistake": Just after Obi Wan kills the two aliens in the Cantina, there are two quick "reaction shots" from both ends of the bar. If you are quick, you can see that they are both shots of the same end of the bar with one shot flipped. You can see the same human with the light blue/dark blue suit on in both shots.
Revealing: For this one, you will need a letterbox copy that is not the special edition. In the final award scene, there is a long shot of Luke, Han, and Chewie walking through rows and rows of Rebel Troopers. If you look quickly, the troopers on both ends of the screen are cardboard cutouts. They are cropped off in the Pan and Scan version, and digitised in the special edition, so it has to be the right copy to see this mistake.
Revealing: When Gold Leader enters the Death Star trench, there's a shot from his cockpit. When Luke, Wedge and Biggs enter the trench later, the same shot is used again. This can be seen from a flash of light accompanied by the sound of a shot, followed by three green laserbolts, then another flash/shot and more laserbolts.
Continuity: In the original version of the film, when Luke and C-3PO are searching for R2-D2 outside the moisture farm, in the shot where Luke says, "It's too dangerous with all the Sand People around. We'll have to wait until morning", the sky changes colour. This has been fixed in the Special Edition.
Continuity: In the scene where Alderaan is destroyed, the "buns" on each side of Leia's head are positioned higher up than in the previous scene she was, so that her earlobes are visible. When Luke and Han rescue her, her hair is back to normal. She's held prisoner in a cell, so it's not likely that she could or would have changed her hairstyle.
Continuity: When the Rebel fleet is on its way to the Death Star, there's a shot where the fighters are seen from behind as they approach the Death Star. Given its tremendous size, and the rate at which it is growing larger to our view, their rate of approach must be incredible. But when Red Leader calls, "Accelerate to attack speed", when you consider their speed with their surroundings, they are now going much slower.