Is it possible for a human to wake up in a wrong way?
I did a quick search and found some research in this area. Sleep inertia is the technical term for feeling groggy for a while after waking up.

In a review article by Patricia Tassi, Alain Muzet (Sleep inertia. Sleep Medicine Reviews. Volume 4, Issue 4, August 2000, Pages 341–353), they define sleep inertia as


  Sleep inertia is a transitional state of lowered arousal occurring immediately after awakening from sleep and producing a temporary decrement in subsequent performance.


and also says,


  Depending on the studies, it [sleep inertia] can last from 1 min to 4 h. However, in the absence of major sleep deprivation, the duration of sleep inertia rarely exceeds 30 min. 


Same article reviews several studies (first report in 1964) on sleep inertia relating to sleep stages:


  Abrupt awakening during a slow wave sleep (SWS) episode produces more sleep inertia than awakening in stage 1 or 2, REM sleep being intermediate.


So yes, sleep stage seems to correlate with "waking up in the wrong way" as @CactusWoman suggested.