"I was told at uni that sleep is cyclical and that as long as you wake up during the lighter part of the sleep cycle you will feel fresh and alert. Wake up during the heavy sleep part and you will be a like a bear with a sore head all day."
-- John D, Glasgow, UK (from the comments of the article)
I have found this to be the case. If I wake up alert, I find it much easier to dive into a problem, which then keeps me awake.