The answer is simpler than you might think: swords did get dinged up and rendered useless in combat. Medieval weapons, while sometimes expensive and valuable, were to some degree considered disposable. If a sword broke or wore out through use, then it was disposed of. After the Battle of Poitiers, which was a rather brutal fight where there was significant amounts of hand-to-hand combat, men who had been issued swords by the king's armory at the Tower of London returned their weapons, many of which were useless or broken. The personnel of the tower picked out the hundred best of the lot and disposed of the rest. Lances, poleaxes, and whatever a soldier might have carried were also disposable. There are accounts of mounted troops being ordered to dismount and cutting their lances shorter to fight more effectively on foot. The medieval soldier, whether aristocratic or not, was a much more practical sort of warrior than modern depictions might lead you to believe.
A pitched battle in the medieval period was a relatively rare event and might only occur at the end of a large and strategically important campaign. The bulk of a soldier's service wouldn't be clashing with the enemy in huge battles in the open field. It would more often be filled with raids, brief skirmishes with equally small enemy units, and waiting/working at sieges. While there might be hand-to-hand combat at some of these actions, it would be to a lesser degree than a full-on battle. Even in the midst of one of the larger battles, combat might take place in the form of smaller clashes over the course of they day rather than one massive brawl where people were smashing at each other for hours. A warrior whose sword had broken might have a chance to grab another sword or poleaxe or whatever off the ground in between clashes.
As an aside, we don't have many detailed personal accounts of medieval hand-to-hand fighting, but some writing from other eras also points to the somewhat disposable nature of the sword. British officers and cavalrymen in India frequently found their swords breaking during combat. It wasn't didn't happen all the time, but even when cutting through an unarmored person, a sword might break on the skull or get stuck in a body. Check out Kinsley's Swordsmen of the British Empire for a huge collection of anecdotes about swordfighting in the age of British imperialism.
Hi, not a direct answer for you, but your question reminded me of another post on sword longevity & maintenance, which I thought you might find interesting
Some other things to think about in addition to /u/MI13 response about sustainability and battle conditions.
A) swords were mostly sidearms throughout history, being used as a backup for a lance, spear or polearm. Those weapons feel the effects of dulling far less than a weapon performing a slice or draw cut.
B) swords can still be effective after blunting, and wouldn't be rendered unusable after minutes of combat barring catastrophic failure. There's been a good amount of testing of historical recreations to learn the properties of swords and how they hold up, and most can be seen online. Just be cautious to make sure the blade being tested is faithful to historical standards, not some cold steel advertisement demo where they chop a car frame in half.
This has actually been a subject of much debate. One of the simplest answers to your question is that armies typically had blacksmiths in their camp who could repair nicks and dings in gear. In that way weapons and armor had a bit of "reuse value." Of course, this could only get you so far. A blacksmith isn't going to be able to repair a broken blade without going through the process of melting down the metal and reforging a whole new weapon - much more laborious and skill-demanding than fixing a dent.
Perhaps the most hotly debated area in this small field is the physical care that one took for their weapon during battle. A modern recreation group called ARMA has proposed a theory that swords, for example, were only ever used to block with the flat of the blade, if at all. This would preserve the comparatively delicate blade and provides much more use out of the weapon. While the evidence is no where near as conclusive as ARMA makes it out to be (my subjective opinion, I can elaborate if asked), I think they bring up a very good concept: the level of protective care taken by the wielder of their weapon. In an era where swords could be relatively costly, it makes sense that soldiers had an incentive to protect not only themselves but their gear as well. Because of all of this, I think that the idea of medieval armies pounding their swords into dull rods against each other's armor is more than a little inaccurate. Just another way that society's perception of medieval martial combat is skewed from Hollywood ;)