Question: I was just reading an e-book and saw this sentence which really confused me:

[![enter image description here](https://i.stack.imgur.com/CuYCV.png)](https://i.stack.imgur.com/CuYCV.png)

"Played twice as fast as written" - This isn't what cut time actually means right, or are there situations where cut time means exactly that?

I thought cut time is basically just 2/2 time signature, but the speed would still depend on the bpm and the note durations...

Answer: 
The e-book was trying to contrast two different meters, 2/2 to 4/4, kind of assuming that the reader might not notice or understand the difference at all. They were trying to provide a simplified first explanation of an unfamiliar meter symbol.

They meant "Crossing the C has the effect that the conductor's hand will start moving in cycles of two beats rather than four beats and you players will have to play twice faster to keep up." 

Except that there's no conductor around which makes the entire difference more abstract to formulate and less constrained to execute.
Adding to the other answers, I played in a group for a few years that liked to take music "a bit" faster than normal, for stylistic purposes, and we would often count 4/4 music in 2 (or 3/4 music in 1) because it was just easier for our leader to count off the tempo (from the piano, often by nodding) when counting at a slower pace and not give herself whiplash. But even if we were counting 4/4 in two, we still played it as if in 4/4 (giving it 4 beats per measure instead of two).

Of cousre, this was the same band leader that would ask us to play in the style of "bad 80's rock music", so perhaps she wasn't always giving us instructions in the most formal music theory lingo possible (although she did know her theory as well).

The problem I have with "played twice as fast as written" is that it ignores the stylistic difference of 4/4 played twice as fast as 2/2. For beginners, this may not matter as much, but it is still there, and we shouldn't confuse them.