It's certainly possible, as such events have been reliably recorded in modern war. For example, from the Victoria Cross citation for Fazal Din:

Naik Fazal Din went to the Bren gunner's assistance immediately but, in doing so, was run through the chest by the officer, the sword point appearing through his back.

• the full citation is available at https://en.wikipedia.org/wiki/Fazal_Din

Similar running-through is also reported in modern murders and accidents.

How likely it is to happen will depend on the path of the sword through the body (bones can stop the blade), and clothing and armour. Thick clothing (e.g., heavy coats worn by Russian cavalry in the Crimean War) has been reported as stopping sword thrusts, and armour will usually stop sword thrusts very well. (The running-through of an armoured soldier, through the torso and two layers of armour, as often shown in movies, generally won't happen.)

Much of the body offers little resistance to a sharp sword. There isn't much in the way of good quantitative measurements on this - stabbing experiments tend to focus on penetration of armour (modern body armour), or knife stabs. Experiments are sometimes performed using ballistic gelatin, which is designed to simulate tissue for gunshot experiments and might not be as good a tissue-simulant for blades. One example of a good ballistic gel experiment, on Pleistocene spears rather than swords, is:

• Wilkins J, Schoville BJ, Brown KS (2014) An Experimental Investigation of the Functional Hypothesis and Evolutionary Advantage of Stone-Tipped Spears. PLOS ONE 9(8): e104514. https://doi.org/10.1371/journal.pone.0104514

wherein it is found that a wooden-tipped spear can be thrust 20-25cm into the simulated tissue. A sword, with sharp edge and much smaller cross-sectional area, will penetrate better.

There are many videos online, on youtube and elsewhere, showing/testing the effectiveness of swords on relevant targets (e.g., pig carcasses, chunks of meat, etc.). These usually feature cutting tests/demonstrations rather than thrusts. However, some show thrusts. The following advertising video by Cold Steel shows sword thrusts through a bag of meat. WARNING 1: don't view if pieces of meat getting cut with swords might upset you. WARNING 2: Cold Steel's style of advertising upsets some people regardless of what is getting cut. WARNING 3: if you follow links from this video to other ones, some will feature pig carcasses etc. being cut up.

• https://youtu.be/-k6OX6XVt6c?t=28

In summary, when it comes to sharp pieces of steel, the human body is a delicate thing. There is a reason why soldiers often wore armour.