The problem that Henry II and his sons faced was not unique. The fact is that any medieval monarch ruling over a large amount of territory (the Angevin Empire was the largest state under a single ruler that Western Europe had known since the time of Charlemagne and Louis the Pious - more about them later) who had a relatively long reign (35 years, well above average for the 19 kings that ruled England in the post-conquest medieval period, 1066 - 1509) during which he had many sons who survived to adulthood many decades before his reign was over (Henry II of course had four) was placing himself in a risky situation. There are many cases of this, which I will explain.

The Carolingians faced this problem on two occasions. The first case would be that of Charlemagne. Charlemagne lived for 72 years (assuming 742 was his year of birth as opposed to 748 as some other historians claim, which would make him 66 when he died so still relatively old) and from his second wife, Hildegard von Vinzgouw, fathered three legitimate sons - Charles the younger (born 772), Louis the Pious (born 778) and Pepin of Italy (born 781) - all of whom were grown men by the time of his coronation as Roman Emperor in the West in 800. With the exception of the revolt of Pepin the Hunchback (born 769) in 792, a son by one of Charlemagne's many concubines who was not considered as one of his heirs (the reason for his disinheritance is debated by many historians, Charlemagne had no dynastic problems after he assumed control of the entire Frankish kingdom in 771 following the early death of his brother Carloman with whom he was originally going to share it. I would argue that this came down to two factors. The first was that the Empire was constantly expanding up until 804, so there was plenty of opportunities for his sons to win glory for themselves i.e. Pepin with the great raid on the Avar hring (their central fortress and treasure hoard) on the Pannonian plain in 795, which was then followed by the disintegration of the Avar Khaganate the following year, Louis the Pious with the conquest of Barcelona in 801, which pushed the frontier with Muslim Spain well beyond the Pyrenees, and Charles the Younger with his various expeditions to subdue the Saxons and Bretons. The second was that Charlemagne, after his coronation as Emperor in 800, subdivided his empire into kingdoms for his sons to govern on his behalf - Charles the Younger got Francia (modern day France North of the Loire, the Low Countries and Germany), Louis got Aquitaine (France South of the Loire and Catalonia) and Pepin got Italy and in 806 devised a plan to divide the empire after his death between his three sons, though this didn't come into fruition when he died in 814 because in the end only Louis outlived him.

Louis the Pious (reigned 814 - 840) had it a lot more difficult. By this point the Carolingian Empire was not expanding any more - military activity was now confined to punitive raids over the frontiers as there wasn't anything left that was feasible to conquer/ worth conquering - and his domestic authority received a severe blow after he was forced to do public penance in 822 after blinding his nephew, Bernard (son of Pepin of Italy), for rebelling against him - you can see the parallels between that and Henry II with Thomas Becket's murder in 1170. Louis had four sons who reached adulthood before his death - Lothar (795 - 855), Pepin (797 - 838) and Louis (806 - 876) from his first wife and Charles (823 - 877). Twice in the 830s he fought civil wars with his eldest three sons over the provisions for succession, and after his death the three surviving sons fought the brutal bruderkrieg which led to the empire being divided into three at the treaty of Verdun in 843. Carolingian dynastic problems did not end there, but from them on it tended to be heirs being minors/ a lack of heirs rather than a surplus of sons. Indeed, Janet Nelson, one of the foremost experts on the Carolingians in Anglosphere academia, has drawn direct parallels between Louis the Pious and Henry II - see her essay "The last years of Louis the Pious" in "Charlemagne's heir" (1990) edited by Peter Godman and Roger Collins.

The next example that comes to mind would be that of the Norman Empire which William the Conqueror created with his conquests of England and Maine (the French region between Normandy and the Loire river) in the 1060s. William had three sons who outlived him, all of whom had reached adulthood by the time of his death in 1087 - Robert Curthose (born 1051), William Rufus (born 1056) and Henry (born 1068). From what Orderic Vitalis says in his "Ecclesiastical History", our most detailed primary narrative source for the period 1066 - 1135, William had a very good relationship with his two younger sons, but a rather bad relationship with his eldest, Robert. When William was about to set off for the Hastings campaign, he named Robert as his heir to the Duchy of Normandy and through Robert's betrothal to Margaret of Maine when he was only 12, which never actually resulted in marriage due to Margaret's death, he also had a claim to Maine, but when he asked his father to allow him to take direct possession of the Duchy of Normandy and the County of Maine in the 1070s, his father refused telling him that he could only do so after his death and Robert, according to Orderic, took great offense at this. Then, when Robert's little brothers played a prank on him, pouring a bucket of water on his head from the gallery of the manor house in which they were staying in, Robert's mates (the various scions of Norman baronial families who had trained to be knights with him), according to Orderic, told him to stand up for himself and so Robert got into a fist-fight with his brothers, with his dad intervening on their side. Robert got in a strop and so ran away during the night with his friends and followers and attempted a coup in Normandy, which was thwarted thanks to William's castellan at Caen finding out in advance, and Robert was forced to go wandering in exile until he teamed up with his cousin Philip of France, who gave him the castle of Gerberoy. King William was furious at this and tried to lay siege to his sons castle, with barely any success in 1077. Robert was reconciled with his father no later than 1080, according to Orderic largely thanks to the input of Queen Matilda who as mother and wife could act as mediator between father and son, but by the time of Matilda's death in November 1083 Robert had fallen out with his father and went into exile again - compare Queen Matilda to Eleanor of Aquitaine, who was accused of goading her sons on in their rebellions against against Henry II (whether those accusations are true or just part of the general Black Legend of Eleanor of Aquitaine I'm not sure). When King William the Conqueror died in 1087, Robert was given Normandy, William England and Henry 3000 pounds of silver - Orderic claims that Henry complained about this, to which William basically replied "your time will come" (this dialogue was probably invented by Orderic, who knew with hindsight that Henry would become King in 1100). Two civil wars took place between William the Conqueror's sons in the period between William's death in 1087 and the Battle of Tinchebrai in 1106, after which all of the Norman Empire was firmly in Henry I's hands.

So basically, Henry II's problem was far from unique and I wouldn't say in either Henry II's case or the others I mentioned it came down to the princes being spoiled brats, not least because attempts to psychoanalyse long dead people are inevitably ill-fated from the outset. Rather, I'd argue that its better look at it in terms of finding an equitable way of providing inheritances for your sons from your large territorial domains when you still don't have clearcut succession laws, and keeping your adult sons busy.