I was using a Fitbit Charge (the one before the just released one). One night I was awake looking at my phone from 1-3am and it said I was asleep. Took it off and put it into a drawer right then.
I work for Fitbit. I don't speak for Fitbit, but in my opinion you should expect it to be the most accurate sleep tracker, and if it isn't you should contact customer service.
Thanks for the advice.