While I can't answer as to how common waging war against one another was, there are definitely examples of nobles waging war without their monarchs. In fact, one of the most famous wars of the Middle Ages, the First Crusade, is a prime example of this. While the Second and Third Crusades would be led by kings, the leaders of the First Crusade were not monarchs. Raymond IV, Count of Toulouse, was probably the wealthiest and most powerful person to go on the First Crusade, though this does not mean that the other crusaders followed him - many different crusade leaders pursued their own agendas as there was no overall military command for the Crusade. Other prominent lords to go on the Crusade included Prince Bohemond of Taranto, Lord Godfrey of Bouillon (accompanied by his younger brother Baldwin) and Robert II, Count of Flanders. Some of these names may be familiar, as some of these men would go on to create and rule the Crusader States - Bohemond became Prince of Antioch (much to the annoyance of the Byzantines), Baldwin became Count of Edessa, and and Godfrey became the ruler of the Kingdom of Jerusalem (I say ruler instead of "King" because he never claimed the title - the first King was his brother Baldwin, who was crowned after Godfrey's death). Raymond of Toulouse technically became Count of Tripoli, though he actually died whilst besieging the city.

Granted, the First Crusade was a pretty exceptional event at the time, as it was sanctioned by the Pope. The First Crusade therefore had a significant amount of legitimacy, even though no kings participated in it.