It depends on which type of armor we're going off of. Mail or gambeson might be penetrated or split by powerful bows and crossbows, which is why shields were the primary protection against missile attacks before full harnesses of plate became more and more common. As armor developed into more sophisticated plate, the need for a shield lessened to protect against those missiles, which opened up the knights to use two handed weapons for greater killing power while not worrying so much about their safety. Part of the reason why plate was so effective against missile attacks is that it was not the sole armor that the knight wore. Most knights had surcoats overtop of their plate, then mail or gambeson beneath, for padding and additional protection. As such, even if a close range crossbow bolt or arrow managed to punch through the plate, it would be unlikely to actually penetrate secondary armor and reach the body within to cause significant damage.

That's not to say missiles weren't a danger to the fully plate armored knight. They could still kill the mount from beneath them, or strike an exposed face if the knight has lifted their visor to breathe or survey the area, as demonstrated at Agincourt. But an actual shot to the breastplate wouldn't likely be too high on the knight's list of concerns.

As for two knights encountering one another on the field, well, again, the situation would be vastly different comparing an early Norman Knight to their more well known incarnation of obscuring steel. As technology increased, so too did the concept of the knight, turning from a land owning, self sustaining warrior, to a position of prestige and nobility, with years of dedicated training behind it. A later era of knight would have years of training, not just in weapon combat, but also stamina building activities such as swimming and dancing, and perhaps most importantly for this scenario, unarmed wrestling.

First, however, to answer your question, would a knight be able to penetrate another's armor with his sword? Not plate and if he managed, it would not penetrate enough to likely injure his foe in any great capacity. For that reason, the sword (for this example I am using the two handed longsword) would not be the primary weapon for the knight, in almost any circumstances. Armor is shockingly effective at stopping slashing and the design and strength of plated armor was extremely strong against stabbing as well. Late knights would favor weapons like the poleaxe if dismounted, for the sheer versatility it would offer, being able to dispatch armored and unarmored, as well as mounted foes alike. However, the longsword would make for an excellent secondary weapon, for situations where the knight could no longer use the poleaxe effectively, or was disarmed. The longsword did not weigh an extreme amount, which made it ideal for keeping as a backup.

If two knights did meet with only their longswords and in full armor, they would of course use it, and there were several techniques designed to use the weapon against plated foes, but it certainly does not look like a sword fight in movies. Half-swording as a technique where the knight gripped the blade of his sword, as the name would imply, halfway up, and use the better control offered by holding his sword in such a way to try and thrust the sword through the vulnerable gaps in plated armor. Another technique, cheerfully named the murder stroke or murder strike would see the knight gripping the sword by the blade with both hands and swinging the hilt, guard and pommel of the longsword like an impromptu warhammer to try and stove in armor and concuss their enemy by bashing them. While the longsword is not shockingly heavy, most of its weight is congregated in the handle (thanks to the weighted pommel) and not the blade.

But perhaps most common of all was the simple act of wrestling their foe to the ground and opening their visor. That could lead to the enemy knight yielding, which would allow the victor to claim them as captive and ransom them after the battle for a reward. If not, a quick dagger thrust through the open visor ended the fight.

In short, armor was extremely, extremely effective in the later stages of its design.