There's a technological development in place in Medieval Western Europe. People moved from chainmail to coat of plates reinforced by chainmail with added gauntlets, banded armour for legs and arms and so on. By the late 14th century, gothic plate armour had developed, and was of such high quality that the shield became superfluous.

For example, it seems like most of the Gotlander peasant militia that badly lost the Battle of Visby 1361 to Frisian and German mercenaries and Danish knights under command of Danish King Valdemar Atterdag wore coat of plate armour and were killed lying down after being wounded in the arms and legs - helmets were removed and a killing blow struck to the head in most cases - showing that even before gothic plate armour, it was hard to get through good quality torso armour.

During the siege of Constantinople 1453, the plate-armoured Genoese mercenaries in service of the Eastern Roman Empire held the breach in the wall by lining up and having the ranks behind them propping them up, making sure they did not fall over - despite vast numeral superiority, the Ottoman forces including their famed Sipahis and Janissaries could not get through this line until the commander of the mercenaries, Guistiani, was wounded and carried away, at which point the morale of the rest of the men faltered and the Ottomans could break through.

Knights and other well-off people who could afford plate armour and expected to meet others that could afford plate armour in battle started using axes, battle picks, maces and various polearms in order to penetrate their enemies' armour, as swords could no longer do it properly.

See this video of two men re-enacting a German military manual as you can see, knights started using their swords as clubs or polearms, aiming to trip or otherwise bring their opponent out of balance and then deliver a killing blow when he was on the ground and unable to defend himself.

If knights were well-armoured and realised the enemy force would be mostly be equipped with weapons that would have a hard time penetrating their armour, I can very well see them dopping their shields and using their swords two-handed.

Take a look at some period artwork. In particular, the Maciejowski Bible (it goes by several names including "crusader bible" and Morgan bible). It dates from the mid-13th century, when mail and shields were still exceedingly common (as opposed to the later shift to plate armor).

What you will notice is that while most knights depicted wield lances and one-handed swords, a few wield larger 2-handed weapons including axes and some sort of big chopper or cleaver (no physical examples of this weapon have been found thus far).

If you look at the guys using their weapons in two hands (and the guys just standing around), you'll notice they haven't thrown away their shields, like Orlando Bloom does in Kingdom of Heaven, but are wearing them slung on their backs. Most of the shields at this time, in addition to the enarmes (arm straps), had a longer strap called a guige. This strap was used to hang the shield from the neck or shoulders. When you're not using the shield or need your hands free, you just wear it slung across your side or back.