In a Game of Thrones, after Tyrion Lannister is abducted by Catelyn Stark, the Lannisters start gearing up to go to war with the Tullys. Now, at this point, both the Tullys and the Lannisters are still loyal to the Iron Throne. Now, this isn't the first time something like this has happened. It is mentioned that the Lannisters have gone to war with other houses before. This strikes me as odd. Imagine if the governor of Washington decided to go to war with the governor of Oregon, there's no way that the United States federal government would let that happen. Yet, in a Game of Thrones, the Iron Throne does nothing to intervene. It's almost like it's considered normal for this to happen. I'm curious if this sort of infighting is realistic to a medieval feudalistic society? I'm not talking about Robert's Rebellion or the War of the Five Kings, where rival lords battled for the title of king itself. I'm talking about instances where lords who are vassals to the same king go to war with each other beneath the king, if that makes sense.
Yes. I answered a question in another thread just a few days ago regarding the practice of homage that dealt with this situation, although the lord in question was not a king.
Violence among rivals under a common lord is quite common. The problem with your metaphor is that medieval kings do not rule over a nation state like the contemporary United States, wherein the federal government can exercise authority over the state governments in certain respects. Kings rule kingdoms, which for the majority of the Middle Ages are essentially larger conglomerations of the personal ties that bound most members of medieval society together. Importantly, these ties are vertical, meaning that they bind the king to individual vassals, not those vassals to one another. That is not to suggest vassals of a common king did not share friendship, marriage, or other bonds—but these were not the default and could just as easily have a negative quality.
If you are looking for details concerning the conflicts between vassals of a mutually common lord, I could suggest a number of primary sources you could consult. Off the top of my head, I would recommend the initial chapters of Joinville's Vie de Saint Louis, The Murder of Charles the Good (not so much for the murder as for the consequences and events afterward), or the Deeds of the Princes of the Poles. All are available in translations.
Edit: I might add that the practice of violence in medieval Europe is a subject of continued fascination. Is the presence of violence in lieu of other negotiated or bureaucratic means of intervention a flaw? Or a feature? In other words, in violence a check against the medieval world being "civilized" (I am cringing just using the term), or simply a part of a society functioning in a way so different than our own its hard to understand? There are lots of takes on what this means. For Thomas Bisson, the struggle to contain violence by kings and other lords in the twelfth century—and their failure to do so—is a sign of societal crisis, whereas for Paul Hyams violence functions as part of an organized system of feuding that, through an anthropological lens, provides a means for resolution and peace making. Shrug. Take your pick.
Some citations: Bisson, The Crisis of the Twelfth Century (2009); Hyams, Rancor and Reconciliation in Medieval England (2003)