After reading the script with his wife, Barry Sonnenfeld (the director) turned to his wife and said "Tommy Lee Jones". She turned to him and said "Will Smith", and that is how they were cast. This is confirmed by Barry Sonnenfeld on the DVD commentary. See more...
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Continuity: Just before the diner scene, where the two aliens are discussing the galaxy, we see the outside of the Russian restaurant. The neon sign says "haroshaya" meaning "good" all the way across the window, in the next shot inside the restaurant, the "ha", which looks like the English letters "xo", is gone and the sign is centered.
Continuity: When the jeweller leaves his shop, the shop window clearly shows the reflection of the parked exterminator's van. Then you see the van being driven up the cross-street as the jeweller leaves in the taxi. Then Edgar/bug parks the van, goes to the shop, and you can see the same reflection in the jewellery shop window.
Continuity: When the countdown timer for the destruction of Earth appears on the viewscreen, you see it count down to 59:56. There is a cut to three cast members including Will Smith and Tommy Lee Jones for a brief dialogue scene. When it cuts to Will Smith walking away you can see the clock still counting down on the screen in the background. Now it is showing the time of 59:54. The aforementioned dialogue section took noticeably longer than two seconds.
Other: There is a mistake in the French subtitle translation. In the diner when the cook asks Edgar the Bug, "where's Ivan?" Edgar responds, "[I] gave him a break". The translator writing the subtitles must have thought he was saying "GIVE him a break" because the subtitle reads, "laisse-le tranquille" which means, "leave him alone".
Continuity: When J is changing in to his suit, he reaches in to the looker with his right hand and pulls out a hanger with a pair of trousers and a jacket on it. his forearm is bare and he isn't wearing a watch. seconds later when he takes the shoes out of the locker with his left hand, his forearm is now covered by the long sleeve of a white shirt and he has a silver watch on. A few seconds later when he lays the jacket over the bench, both his forearms are bare and again he isn't wearing a watch.
Visible crew/equipment: In the scene in the beginning where Will Smith chases the alien who jumps off the bridge onto the street. Will Smith jumps from the bridge onto a tour bus, then jumps on the back of a newspaper delivery van. In the passenger side of the van is a cameraman with his back towards the windshield in order to film down the side of the van as Will Smith jumps off the van.