I can answer your question regarding the situation specifically in Hungary, as I am most familiar with Hungarian medieval history. I will have to say, that my sources aren't translated to English though.
There were quite a few noble houses in the course of Hungarian history who get incredibly wealthy, and some of them even became royalty.
In the XIth century, with the establishment of the Christian state the king took most of the land into his own possession. Most of it was governed by appointed royal officials. While the offices themselves did not become inheritable as it happened in some other medieval states, after a while kings started to reward their loyal retainers with lands which could be inherited. That's how the first aristocratic dynasties emerged.
The royal power was absolutely overwhelming throughout the first two hundred years, but things started to go downhill by the end of the XIIth century.
II. András (1205-1235) was a troublemaker in his youth, often taking up arms against his older brother (Imre, 1196-1204), and after he became king following his nephew's (II. László, 1204-1205) untimely death, he stirred up even more trouble. He really liked wars, and to secure support for his foreign escapades he gave out lands like candy. It backfired though, because the increasingly powerful nobles grew tired of his arbitrary ways, and forced him to acknowledge checks on royal power. In 1222 he was forced to give out the Golden Bull, a charter of rights similar to the Magna Charta.
The son of II. András, IV. Béla (1235-1270) was the polar opposite of his father, he believed in a strong, centralised state, and wished to rule without checks and balances. He took back donations of his father and arrested nobles who questioned his authority, or who have wronged him in the past, blatantly disregarding the Golden Bull. However in 1241 the Mongols invaded Hungary and the royal army suffered a crushing defeat. At the same time some wealthy nobles who built up-to-date castles were able to hold out and survive the one year long Mongolian occupation. This convinced Béla to turn upside down his former policy, and now he was giving out lands with the condition that new owners build castles on them.
This dealt a crushing blow to the royal power. While the network of castles built up all over the country worked well against Mongolians when they came back in 1285, castles were mainly used now to oppose the king. Whenever the king wanted to lecture an aristocrat, the troublesome nobles just shut themselves into their castles, and waited till the king got tired, or trouble reared up its head somewhere else.
In 1301 the male line of House Árpád died out, and by this point the country became a very fragmented entity, where oligarchs (prominent families were Csák, Köszegi, Aba, Borsa, Ákos, Babonić and some others) were ruling their own parts with almost absolute power. It took two decades to firmly establish a new king, and break the power of the oligarchs.
The cycle of centralisation and decentralisation then took place again, but this time it took less than a century. I. Károly (1310-1342) broke the power of oligarchs, but his son, I. Lajos (1342-1382) had an aversion against sensible domestic policies, and was fond of squandering blood and treasure in fruitless military escapades. Again magnates were established, and again they used their power to force more concessions from the king.
With Zsigmond (1387-1437) things became even worse. First, he had to deal with a powerful and aggressive neighbour, the Ottomans, a problem the country did not have for the last hundred years. Zsigmond was really bad with financial matters, and to finance the military he pawned a lot of royal property to nobles, or outright sold it.
The most prominent family rising up under Zsigmond, and the kings following him were of course the Hunyadis, the son of Hunyadi János, Mátyás even became a king in 1458. Funnily, I. Mátyás (1458-1490) was one of the most effective king when it comes to centralising, and breaking the power of nobility, but he was never able to concentrate all his efforts on domestic policy, mostly because he too was fond of the idea of becoming a Czech king and ultimately Holy Roman Emperor as well.
After the death of Mátyás, I. Ulászló (1490-1516) was elected, realising the Czech-Hungarian personal union but not exactly in the way Mátyás wanted. Ulászló was unable to firmly establish his power, and in the end the Szapolyai family became so strong under his and his son's rule that after 1526 they were able to imitate the Hunyadi family and get themselves into the throne.
The next decades were that of bloody and endless civil war, with the Ottomans occasionally showing up and thrashing everybody. Szapolyai János tried to establish himself as an Ottoman client king, while I. (Habsburg) Ferdinand asserted his rights as the legal heir of II. Lajos. When finally it was sorted out, and an agreement was signed between Ferdinand and János, János kicked the bucket, and it started all over again. In 1570, after half a century of bloodshed the de facto partition of the country was acknowledged by the Szapolyai family and the Habsburgs - II. János accepted the Maximilian I. as his kings king, in return Maximilian gave János Transylvania and the Partium as inheritable fiefdom. Of course, János died shortly without an heir, which caused all kinds of troubles again, and lead to Transylvania seceding from the Kingdom of Hungary for the next hundred years.
To be continued