There are a lot of different factors/reasons for how one wakes up. For example, if you are deep in emotion, lets say guilt or shame, all day yesterday and when you wake up, that emotion is still there and it weighs you down. Another factor can be physical strain, like you worked all day yesterday 14 hours, and you had little breaks or rest and you ate very little. That can also lead to feeling tired waking up the next day. So just like anything in process, your experience of yourself can be affected by who you are in your mind level, and what you body goes through on a physical level. As you can imagine, and perhaps have already lived some times: where you are stable physically, and mentally/emotionally... you may have some morning where you wake up fine. But at the same time, process right now is not meant to be a stable smooth ride, because that is not our reality both as who we are and in physical reality. There will be unexpected points, as well as personal processes that need to be walked. This will have affects on how you sleep.
So to answer your question, it does exist.
Now, a question that I see may be hidden within your question is whether having FEELINGS and positive experience lead to an experience of waking up rested. Yes, I have experience that. Where its a positive part of the mind that makes ones feel good. This isn't something stable of course because its based on the mind.
Now, another question that a person could have here is whether you can somehow live/create waking up rested. So there is an element here of where you decide who you are in the moment. That always exist in every moment. So you can will yourself, move yourself, as a kind of positivity. This here you can make it practical by using words that work for you. For me the word embrace the moment, embrace the day, and embrace myself supports in the waking up process. Also even pretending I only have today to LIVE. So that supports within waking up WELL. I would not focus on waking up RESTED. I would focus on living waking up WELL. Because you can't always control how rested you were during sleep, because maybe you are forced to wake up early or forced to sleep late, but you can control HOW you wake up through self-movement.