As with most questions, the correct answer is "it depends". Context of use matters quite a bit when you are talking about the physics of these blade actions. Certain types of fencing will not lend themselves to the edge becoming chewed up and scored as Hollywood and videogames might lead us to believe. For example smallsword blades (which were used during war) are often edgeless, and and only used for thrusting.

Similarly a cut oriented sword, like a sabre, could still be going up against spears, shields, halberds, maces, helmets, mailed targets, or even plates of armor...any number of items which could cause damage to the blade, but not cause the 'saw like' effect that you perhaps have in your mind.

Also, you must consider the type of fencing style. For example Chinese Jian will often show large 'cuts' from opposing another swordsman, but because of the type of fencing that is use, the Jian will take the blow on the flat spine of the blade instead of the edge. This can be seen on this Ming example, with two large cuts in the middle of the blade.

There are many examples of blades in museums with extensive edge damage. Some even with an exaggerated saw-like edge. However it is very difficult to determine how these damages came to be inflicted on any given sword unless there is a direct accompanying account explaining the infraction. A chip or nick could have been caused by a sword, knife, rock, and could have happened during battle, training, or anytime over the course of the sword's life. Often times swords were just 'played with' either by their original owners, or in the decades and centuries which followed. Experimental research such as that done by Raphael Hermann can sometimes help explain what is likely to have cause certain types of damage on blades, but again, there will never be 100% certainty.

There are some historical accounts discussing how swords became 'notched' in use, and would need to be repaired or resharpened to be effective again:

From the late 12th century

“The condition of the swords is not improved, nor that of the helmets and shields, which are dented and split; and the edges of the swords are nicked and dulled. For they strike each other violently, not with the flat of the swords, but with the edge.”
(Yvain or, The Knight with the Lion, by the French poet, Chretien deTroyes Part III: Vv. 4635-Vv. 6818)

From the 13th Century:

“Tryggveson’s men, he observed with surprise, were striking violently on Eric’s; but to no purpose: nobody fell. ‘How is this?’ asked Tryggveson. ‘Our swords are notched and blunted, King; they do not cut.’ Olaf stept down to his arm-chest [and] delivered out new swords.” (Early Kings of Norway. Chapter III, Thomas Carlyle. Volume 19.)

And finally from the 15th century

“the blows fell upon good armour, though not so good but that it was broken and bent in many places. The sword he used was like a saw*, toothed in great notches, the hilt twisted by dint of striking mighty blows, and all dyed in blood.” (*Chronicle of Don Pero Niño against the Moors)

References:

Bronze Age Swordsmanship: New Insights from Experiments and Wear Analysis Raphael Hermann December 2020 Journal of Archaeological Method and Theory 27(328)