Continuity mistake: The brain that Frankenstein draws on the blackboard in the first scene is different from the one behind him as he is seated and lecturing. Check the shading on the brain stem. (00:05:55 - 00:09:35)
Revealing mistake: In the classroom Frederick stabs himself in the leg with a scalpel. Look closely and you can see the thick pad covering his thigh inside his trouser leg. (00:11:10)
Continuity mistake: During Inspector Kemp's visit at the Frankenstein manor, Frankenstein is playing darts while he is having his conversation with Kemp. He throws the first dart at the bottom 10, but by the time he throws the second dart at 7, the first dart he threw at 10 is now at the bottom of 18. (00:56:20)
Chosen answer: While no specific date is given, Mel Brooks wanted to do a tribute parody to the classic 1930's horror film, thus the Black and White film and the 30's style film techniques. However, Hans Delbrück (the brain Igor is suppose to steal) was a real-life person who died in 1929. Given that and the cars seen, it's a safe bet it's suppose to be set in the early 1930's.
Bishop73