The entire Middle Ages was a continual power struggle between the church and the nobility. The church was constantly trying to keep power over the nobles, and the feudal structure was an unstable and sometimes uneasy balance between the wild and unpredictable warrior-elites (who became knights) and the much more tame but often more corrupt church hierarchy. Both sides needed the peasantry on their side, and the peasantry would favor whichever elite gave them more advantages.
The church created useful mechanisms like allocating a massive amount of penance as a requirement for each life a warrior might have taken in battle in order for that warrior to get to heaven. Thus, if a knight was called up and answered to his oath of fealty, he was going to get "in the red" with the church pretty fast if he had to go into battle. Don't worry, sir knight (who now owes 300 years worth of penance)! We have an entire monastery worth of monks who will happily do that penance on your behalf! All you have to do is pay for their upkeep.
This balance went back and forth continually through the entire Middle Ages. The famous battle of King Henry VIII with the pope stemmed from the restrictions the church put on divorce, but what most people don't know is that there was a very good reason that Henry had a good reason to need to marry so many wives (and obviously divorce them to marry more). The laws of "Casus Belli" (just war) required certain conditions in order to lawfully invade another kingdom/fief, etc (obviously Christian ones). One of the principal just war conditions was a claim that a noble had inherited on a particular fief. You could go to war on behalf of a spouse, even if you didn't have the claim yourself, which meant that Henry was strategically picking spouses and marrying them in order to gain claims on land that he needed. Note, I said needed, not wanted, because Henry's domain was basically indefensible and illogical when he came to power. He was attempting to create a geographically logical area that would work as a kingdom and could be defended.
Why was there such a thing as "casus belli?" Because the CHURCH wanted to limit the power of the savage knights!
So what happens if you have a pope who absolutely will not cooperate or a church that goes too far? You create an "antipope". You basically just set up an alternate church hierarchy yourself, which happened many times through the entire Middle Ages. The most famous example of this (and the cleverest) was the creation of the Anglican Church (clever because instead of a "pope", the church used an "Archbishop", which caused a lot less political turbulence and allowed it to continue to operate right alongside the rest of Catholic Europe). There were many "anti popes" and even cases of multiple, basically legitimate "popes" who were operating at the same time, both trying to control the same church.
So what do you do if you have a church split into a thousand pieces by the centripetal power of lots of local nobles? You declare some Heresies! Declare local factions to be heretical and purge them (basically force them back into the fold). If THAT doesn't work, you might eventually have to even call an Inquisition down on them (that was basically the later part of the era when the Catholic Church was getting very serious about centralizing power over all Christian lands in Europe).
Basically, the feudal system had a natural balance of power going on. If one party became too tyrannical or went too far, the other side would do something drastic (like set up an anti pope or declare a heresy). The power went back and forth, but neither side ever had a total monopoly. It was really the peasants who held many of the cards, and they needed the nobles to protect them physically (nobles were the military and police) and the church to regulate society and provide a lot of the identity that modern societies place in nation-states. Basically, because they were the ultimate arbiters in the never ending fight for dominance between the two elite groups, the peasants enjoyed a LOT more rights and freedoms than people today realize. They could (and DID) take kings to court over land disputes, for example.
What eventually destroyed the power of the peasants was the collusion of the church with the top level monarchs in Europe to destroy the power of the lower level nobility and seize their lands. This evolved into the imperial system that we are more familiar with from the Renaissance era (and which led to the French Revolution among other things because it was very tyrannical). Basically, what people don't realize is that the "awful" "dark ages" were better for COMMON PEOPLE than our much celebrated "renaissance".
To get back to your original question: if you are basing your story on anything similar to historical Europe and "Christendom", then almost anything you can imagine as a major conflict between nobles and clergy isn't just possible, but it probably happened already. (I didn't even get into the religious wars period, which might be even more in-keeping with your story theme. During the wars following the Reformation and counter-reformation, almost HALF of the population of Germany was annihilated!)
You might want to look up some of the attempts to create Utopian societies that came out of the reformation and some of the bloody history that followed during the religious wars period for atmosphere.