Although the title of your question refers specifically to the 12th century, your question asks for the High Middle Ages which is a longer period. As such, I will focus on the High Middle Ages of France. There are numerous examples of how succession might work during that time period thanks to the multitude of succession crises of France and Burgundy. Overall, primogeniture was not the only law at play during that time period. Proximity of blood was also taken into consideration. From the Wikipedia article on the Duchy of Burgundy some cases of these 2 laws together are mentioned:
The Duchy of Burgundy, however, proved a greater challenge to jurists. In the duchy, as in much of Europe at this time, two principles of inheritance were held valid: that of primogeniture and that of proximity of blood. A case of primogeniture was the succession of the English crown in 1377, which at the death of Edward III was inherited by his grandson Richard, the eldest son of his deceased eldest son Edward, rather than by his son John of Gaunt, the eldest of Edward III's sons still living. A case of proximity of blood was that of Artois in 1302, which had on the death of Count Robert II been inherited by Mahaut, his eldest living daughter, rather than by his grandson Robert, the eldest son of the count's already deceased son. In some cases, the two principles were able to mesh together: in the case of Boulogne and Auvergne, for example, John was the second son of Robert of Auvergne, Philip's great-grandfather, and the nearest ancestor to Philip to have surviving lines of descent following Philip's death. John was therefore both the most senior heir to Robert following Philip's death and also the closest to Robert by descent. In the same manner, Margaret of France was the closest heir by both primogeniture and proximity to her mother, Joan of Châlons, Countess of Burgundy and Artois, Philip's great-grandmother and, again, the nearest ancestor of Philip to have lines of descent surviving the Duke's death.
Therefore the jurists that had to decide on the validity of the claims of potential successors had to take all these into account, and as was demonstrated during the Hundred Years War, the decision could be based on other external factors on arguable shaky legal grounds, such as the sudden revival of the salic law as demonstrated here:
By proximity of blood, the nearest male relative of Charles IV was his nephew, Edward III of England. Edward was the son of Isabella, the sister of the dead Charles IV, but the question arose whether she should be able to transmit a right to inherit that she did not herself possess. The French nobility, moreover, baulked at the prospect of being ruled by Isabelle and her lover Roger Mortimer, who were widely suspected of having murdered the previous English king, Edward II. The assemblies of the French barons and prelates and the University of Paris decided that males who derive their right to inheritance through their mother should be excluded. Thus the nearest heir through male ancestry was Charles IV's first cousin, Philip, Count of Valois, and it was decided that he should be crowned Philip VI. In 1340 the Avignon papacy confirmed that under Salic law males should not be able to inherit through their mothers.
A similar situation arose during the second Burgundian succession crisis:
To the jurists of the duchy, these facts presented something of a difficult legal problem, for the two claims stood more or less equally in terms of justification: Charles II, as the great-grandson of Robert II by his elder daughter, had a superior claim to John II in terms of primogeniture; John II, as the grandson of Robert II by his younger daughter, had a superior claim to Charles II in terms of proximity of blood. Were it simply a legal issue, the King of Navarre would certainly have had as good a chance of inheritance as the King of France, and perhaps better: proximity of blood was beginning to lose force in Europe, and, as events would subsequently prove, Burgundy had no intention of being absorbed into the French royal domain. But there was more in play than a simple legal issue: the Hundred Years' War was in full flow, and the King of Navarre, as an ally of England and an enemy of France, was distasteful to the Burgundians, who in meetings of the Estates during John II's English captivity had been consistently loyal to John and his son the Dauphin, and opposed to the King of Navarre.
Therefore, we can conclude that succession was no simple matter at the time. Parties with valid legal claims could be overlooked in favor of other candidates for reasons of state, such as the avoidance of a future crisis, the protection of state rights etc and ultimately might was right. If you could press your claim through force of arms and gained victory you had the superior claim.