Deliberate mistake: During the 6/11/04 episode, Jax and Courtney are having dinner in Italy. Carly, who is in New York, does some sleuthing, finds out where Jax and Courtney are, and flies to Italy. Miraculously, Carly manges to fly from New York to Italy and arrive while Jax and Courtney are still eating dinner.
Battlefield - S26-E1
Deliberate mistake: Look for the "creases" in the wall inside the TARDIS, which look as if the wall was just a sheet hung up behind the console. (In fact it actually WAS a sheet and not a "solid" wall - the TARDIS wall prop was apparently unusable due to water damage, due to poor storage between seasons. This also explains the lack of TARDIS interior scenes throughout the rest of season 26: it was not thought worth repairing the TARDIS set [or building a new one] when the show was in imminent danger of being cancelled after season 26).
Deliberate mistake: Though by 1964 it was already suspected (and later confirmed by Mariner IV) that Mars has an atmosphere humans can't breathe and temperatures too low for humans to tolerate, Merritt and his crew are here exploring the planet sans spacesuits. This was "fudged" because space helmets A)are expensive, B)reflect cameras, and most importantly, C)obscure the actors' faces. (00:24:00)