Factual error: The bad guy tells the townsfolk to 'Get out of Dodge.' Not a saying used in 1879, despite Dodge City being a real place. In fact by this time Dodge City was no more violent than many other places, so it wouldn't have been used as a direct comparison either. There's no record of this specific phrase being used until Gunsmoke, in 1955.
Stand by Ben - S1-E8
Factual error: The Blur song "Song 2" plays during the intro escape. The episode is set in 1996. The song came out on April 7, 1997.
Factual error: Depeche Mode's "Policy of Truth" plays but the episode is set on October 17, 1989. The song wasn't released until May of the following year.
Factual error: Eugene says he was a bit of a geek in high school. 'Geek' as a term for a socially awkward person wasn't used until the 1970s. In 1962 'geek' still meant a carnival performer who bit the heads off live animals.
Other mistake: When Hannah turns on the magnet, it's so powerful that it pulls the sword out of Donovan's hand from across the room, but no other metal object is affected.
Ben & Teller - S2-E2
Continuity mistake: The hole in the drywall that Ben creates goes from a circle to a square, and back to a circle.
Factual error: The psychiatrist asks Ben who the current president is, and Ben, not knowing it's 1954, guesses JFK. The doctor responds "The senator? He'd love that." But the nickname JFK wasn't used until the 1960 election.
Factual error: Some of the crew members are standing on deck while the ship is firing. They're close to the guns. Not only are the guns very loud and thus would destroy their hearing, those men would potentially be physically injured by the shock.
Revealing mistake: During the police chase, the van runs a stoplight. Tire marks from previous takes are visible on the ground, and the van follows them.
Factual error: At the beginning, during the launch, it's stated that it's 1998. We learn the US astronauts are on the space shuttle Atlantis to deliver the first piece of the International Space Station into orbit. However, Atlantis never had a flight in 1998. The American ISS crew flew on the Endeavor. And the Americans launched the second piece of the ISS. The first piece was launched a few weeks before by the Russians out of Kazakhstan.