Ok, so first thing's first: there was no 'feudal system'. Credits to /u/idjet and /u/TheGreenReaper7.

If anything though, this maybe makes your question even more pertinent, because to modern eyes if there wasn't a 'system' keeping everything in place, it's very easy for things to fall apart and for lords to seize power from the king. And yet, during the central middle ages, Marc Bloch's "second Feudal Age" from 888 to c. 1200, this didn't happen. What we can see, however, is a relative increase in the power and autonomy of the aristocracies of Europe relative to the monarchy, most notably in the (gradual) spread of the idea of heretibilty of offices. What this means is that instead of titles (like count, viscount, duke, marquis, etc.), and the land and power that accompanied them, being distributed by the king (usually for the life of the holder), gradually from the middle of the ninth century it was expected that the previous office-holder's son, or nearest male relative, would inherit. The old-school interpretation of this development is that kings became much weaker as a result, and their nobility too powerful, threatening the very stability of the kingdom (let's call this the 'Over-mighty Vassal' approach).

The problem with the 'Over-mighty Vassal' approach is that it envisages medieval politics as a zero-sum game, a constant struggle for power in which there can be only one winner and one loser (for it almost invariably assumes the aristocrats are a homogeneous group without any conflicting interests, whose only goal is to increase their own power at the expense of the king). If you dig a little deeper into this approach, you'll find an implicit moral judgement, born by the dominance of centralised nation-states in modern Western history: weak kings, who can't control or micro-manage their aristocracies are bad, and failures; strong kings are good, because they help to increase the power of the state and they put those who would stand in the way of 'progress', i.e. the 'over-mighty vassals', in their place. Once we realise that this is a problem, and that these are the assumptions which lie behind our thinking, we can re-formulate your question - Why would the lords want to seize power from the king? And why would the king want to prevent the lords from seizing power?

The reality is that during the central middle ages, the king and his kingdom's aristocrats saw themselves as allies, as partners in ruling the kingdom. There's no evidence that kings wanted to take control of their nobility and micromanage them, nor is there any evidence that those nobles wanted to get rid of the king and split the kingdom into a dozen little pieces. Each party played a particular role in the government of the kingdom. The nobility acted as counselors and agents for the king - he was supposed to consult them regularly, to only make decisions with their approval, they kept order in their spheres of influence, they went on embassies for the king, they ran law courts and collected taxes, and supplied the bulk of the royal army. The king, meanwhile, was vital for the aristocracy - he provided an arbitrator for their disputes, provided patronage for them and for their followers (in particular for church offices) and, most importantly of all, legitimized the power they wielded and the actions they performed using that power. A reflection of this symbiotic system, and it's hard to say whether this is a cause or a consequence of its development, is the fact that by the mid-eleventh century every major European royal family had been merely aristocratic a century before: the Kings of England had been Dukes of Normandy, the Capetian Kings of France had been Dukes of the Franks and the Salian Kings of Germany/German Emperors had been Counts of Worms and Speyer. Kings during this period are best seen as 'first among equals' in relation to the aristocracy, rather than being in opposition to their nobles.

Now, up to now I've been talking in very general terms. Obviously, this is AskHistorians, so the details are not so simple, and there was a huge amount of chronological and geographic variation - this is a broad outline, and things were always changing. But to show you how things worked in practice, I'll give an example in detail, drawn from early 10th-century France. It's an example of what happens when the system I've described above goes wrong, when kings and their aristocracies are in direct opposition, and what their actions can tell us about how they expected the system to work.

This event takes place in Northern France in the years 922-3. The king of France, Charles III (as known as Charles the Simple), was not acting how a king was supposed to act: he was not taking the advice of the leading men of the kingdom and was in fact listening to only one man, a 'mediocre' advisor by the name of Hagano. After trying repeatedly to come to a compromise over the issue, in which after a walk-out at his court in 920 Charles made a big show of taking advice from other nobles as well as Hagano, the nobility had had enough, and rose up in rebellion against Charles, led by Robert of Neutria. Robert was probably the most powerful noble in France at the time; he was Count of Paris, Poitiers, Angers and Orléans and the brother of King Odo, who had ruled France while Charles was a minor. As much as Game of Thrones fans would like it to be, however, this was not just Robert's Rebellion. He was supported by just about every important noble in France, the one big exception being the new kid on the block, Rollo, the ex-Viking count of Rouen. The nobility drove Charles out of his kingdom and crowned Robert as king at Reims in 922.

A year later, however, Charles was back, with an army. Robert met him in battle at Soissons on 15 June 923, where Robert was killed in battle. Despite losing their king, the army of the nobility was victorious and Charles was captured soon after. He would be held prisoner by Count Heribert the Old of Vermandois for the rest of his life. The nobility passed over Robert's son Hugh and installed his son-in-law, Raoul, Duke of Burgundy, as the new king.

There are some important points to take away from this story. Firstly, it was a very rare event that pitted (almost) the entire nobility against the king, and it usually meant very bad news for said king (another example of the same thing is the last few years of King John of England's reign). While it was not at all unusual for a king to argue with some of his nobility, or even to go to war with them, he would always have other nobles on his side, and even more staying neutral and watch from the sidelines. That was how regular politics worked. Situations like in 922-3 were not at all the same, and constituted serious crises. After such a minor war, kings would endeavor to restore some consensus to politics and to bring the loser back into the fold via some sort of compromise. This even happened in 922-3: Rollo the Norman kept all of his lands and power, despite having supported Charles 'til the end; after Raoul's childless death in 936, it was Charles' son (Louis IV), rather than Robert's son Hugh, who succeeded him.

Secondly, the motivation behind the rebellion is all to do with the king not sharing power. The nobilty aren't rebellion because they want more power, they're rebelling because Charles is denying them the power they used to have and which they should have. Charles is supposed to listen to his nobles and to share his power with them, not to give it all to some up-jumped minor aristocrat. It's also significant that the nobility, as well as Charles, seem to have made a significant effort to compromise over the Hagano issue for some time before the rebellion actually broke out, even if it was ultimately unsuccessful. All of these people had a clear idea of how politics was supposed to work, and the failed compromises and the rebellion itself were an effort to get back to the status quo.

The significant acts of the rebels confirm this. They didn't divide the kingdom up and each take a slice. They elected a new king from amongst themselves, a king who would (it is implied) uphold the norms of politics. The fact that after Robert's death, the crown passed first to his son-in-law and then to the son of his enemy, rather than to his own son, who was probably in his mid-twenties at the time of his father's death, shows that this was not a vendetta by Robert, nor an attempt by his family, or the nobility in general, to seize power from the king, as we might expect if we followed the 'Over-mighty Vassal' theory.

So, in summary (here's the TL;DR), even though we can't call it a 'feudal' system, there was a system of unspoken and unwritten rules in the central middle ages which meant that in normal time, the king and the nobility shared power, and supported one another. Despite the fact that conflicts between the king and individuals or factions within the nobility were common, damn-near ubiquitous you could say, neither side saw any advantage in trying to overturn that system, as 'Robert's Rebellion' (to use an inaccurate but fun shorthand) in France in 922-3 shows.