Is it possible to prove what happened?
The answer is yes but the show, like all other books and documentaries fails to do it. Here's how: There are two ways to determine the timing of the shots that were fired. NOVA touches on this but fails like all others do to sort out the details. If you look at almost any source, you will NOT get the right answers. It turns out there are errors that were made during the HSCA investigation that the show only touches on.
Obviously if the time between shots is the same in different methods, it means that both of the methods are valid. In this case, the two methods are the Zapruder film and the audio recording. Let's look at the Zapruder film first.
The first time we can definitely determine a reaction to a shot is Zapruder frame 224 (the show says 225 but either is close enough). And obviously JFK was shot in the head at Z frame 313. The film speed was about 18.3 frames per second. A simple calculation shows the time between those two shots is approximately 4.8 seconds.
Looking at the following figure for the acoustical evidence analysis it ALSO shows that two of the shots were fired 4.8 seconds apart. 
What is the probability that two independent ways to measure the time between shots would show the same time? The time of 4.8 seconds common to the Zapruder film and the audio recording obviously suggests that the acoustical data did capture the sounds of the gunfire in Dealey Plaza. Search the over 1000 books and documentaries and you will not find this fact anywhere.
The acoustical data analysis suggests that shot #4 was the one fired from the right front which is obviously the one that anyone would align with the one that hit JFK in the head at Zapruder frame 313. Looking at the acoustical data, it shows the last shot was fired 0.7 seconds after the JFK fatal head shot. What happens in the Zapruder film 0.7 Seconds AFTER JFK is shot in the head?
These two frames (Z224 and Z227) are about 0.1 seconds apart. How did Connally's head move so fast?
The acoustical evidence helps answer that question. The acoustical analysis used a technique called echo correlation analysis. It gave the timing of the shots but it also gives probabilities of the LOCATION OF THE SHOOTER. Shot #4 was fired from the right front and Shot #5 was the last shot fired from the Texas School Book Depository.
Per the acoustical evidence and the Zapruder film, here is what happened during the last two shots.
Z224(or 5): Connally reacts to a shot
4.8 seconds later
Z313 JFK is shot from behind with a bullet fired from the right front (grassy knoll)
0.7 seconds
Z325 The last (3rd shot from TSBD) shot fired from behind misses JFK's head and hits Connally in the back bending him over.
Here is another reason that makes more sense than what they showed on NOVA:
1. Connally said after he heard the first shot fired he turned to look over his right shoulder, he didn't see anything and as he was turning back toward his left he felt a blow to his back and it bent him over. Connally is NOT bent over until frame 325 when the acoustical evidence indicates the last shot was fired.
As Connally was turning to look to his right, he has no blood on his shirt and he obviously does not look like a man who has a exit wound in his chest the size of a baseball. Where is the blood? How did he turn (before JFK is shot in the head)? The answer is simple: Connally was not shot in the back until a split second AFTER JFK was shot in the head exactly as the acoustical evidence indicates. Not one of the facts presented here was discussed on NOVA. WHY? Because it invalidates virtually everything they presented.
How did Governor Connally turn around in his seat with an exit wound in his chest the size of a baseball? The answer is he didn't, he was not shot in the back until a split second after the JFK fatal head shot.
Continuing on and using the acoustical evidence and the Zapruder film together, it is possible to determine a unique solution. Unique? Nothing else is possible when you look at the two forms of evidence together.
Finding that the acoustical evidence is valid is much more significant than what you might at first think. It proves for example that a conspiracy killed JFK. It proves that the US government forged evidence. It proves that it wasn't just a conspiracy, it was a powerful conspiracy composed of people who could forge evidence and confuse the investigators. When you put this all together, you can finally know WHO and then it quickly leads to WHY.