Okay... so I've been watching a lot of Game of Thrones, and it got me wondering if it was a realistic depiction of kings. Could a king really do as they pleased? I understand that people believed in the "divine right of kings" but were there examples of kings being deposed when they went too far?
The divine right of kings is actually a relatively late invention.
The answer to your questions is that medieval kings were absolute monarchs right up to the point at which they weren't. That is, you can issue all the edicts you want, but if you don't have the power to enforce them, they won't be enforced. The French monarchy spent most of the first two centuries of the second millenium consolidating its immediate power base, for example, before being able to expand it outward. Moreover, every decree you make will probably piss somebody off, so ruling was (and is) a balancing act. Piss off too many people, and you won't live very long.
The best book on the subject, and one of my favorite books period, is:
Absolutism wasn't invented until the 17th century, so that was after the Middle Ages were over. During the Middle Ages, there was generally some sort of division of powers between the king and other princes. For example the Golden Bull of 1356 in Germany or the Magna Charta Libertatum of 1251 in England were laws that limited the rights of the kings with respect to other nobles. The kings of the Holy Roman Empire tried hard to establish a hereditary monarchy during the Middle Ages -- but they ultimately failed; the other princes insisted on their right to elect the king. During the late Middle Ages, many cities managed to become basically independent from royal rule -- they called themselves free cities. Then there was the church: many kings got into conflicts with the Pope, and lost.
So: no. Absolute rule in the Middle Ages is fictional. That said, there was no concept of civil rights either -- the ones who balanced a king's powers were other nobles and not the common people.
Kings varied a lot in how powerful they were - partly due to how rich or poor they were, also how diplomatic they were in dealing with powerful nobles and neighbouring kings, and partly also to how lucky they might be. Coming to the throne as a child (eg several generations of the Stuart kings of Scotland), being defeated in battle (many kings), having a bout of illness or insanity (eg Charles the Mad of France, who believed he was made of glass and lost half of France, and his grandson Henry VI of England who seems to have had delusions) were opportunities for the big aristocracy to enhance their powers or for neighbouring countries to invade. As telkanuru says, they may have been absolute monarchs in theory, but in practice they weren't. English monarchs also depended for part of their income on Parliament, which might or might not be easy to influence or control - which of course leads to the English Civil Wars and a king having his head cut off. Charles I seems honestly to have believed he was king by divine right, but the axeman thought differently.
John Gillingham (a very reputable scholar on kingship, chivalry, and many other aspects of the central middle ages) has made a copy of his paper 'Expectations of Empire: Some twelfth- and early thirteenth-century English views of what their kings could actually do' available on his academia.edu profile.