A quadruple axis is possible

Could the best possible archer in real life get the shots that Katniss Everdeen routinely takes?

Basically, I agree with ShemSeger's answer:

But don't take my word for it, see for yourself:

This guy is a real Hawkeye, he says very factually that he can hit him with a bow and arrow if he can see him. And if you watch the video you will see that it is true, he hits everything including a tiny pill that is thrown into the air.

I would like to add a few things, however.

Shooting style

In the movies we see her using the bow to hunt. That's why she learned a "hunting style" for shooting.

So here comes the part where I disagree with ShemSeger. I don't think Lars Anderson is a good analogy. I want to point out some great shooters who shoot to kill (that is, to hunt). It's the shooting style that best fits the frame of the film.

Here is a list:

Just comment if I forgot someone :)

Conclusion

So the answer to your question is, "Yes, it is absolutely possible (even to top it)." Just check out some of their videos. The accuracy is absolutely fascinating. For example, Jeff also hunts with a 65 pound bow, which is quite a bit. A bow like this one would be perfectly capable of shooting an explosive arrowhead towards an airplane (if it really went down, etc is another question :)).

useful information

The funny thing is that the actor was taught an Olympic style of shooting, which is different from the style of hunting. I don't want to go into these differences as that would be a subject for another question.

This video sums it up pretty well: https://youtu.be/nKS5dDQxp3M

From https://scifi.stackexchange.com/questions/108633/does-katniss-depict-proper-handling-of-a-bow:

Jennifer Lawrence was trained by Olympic archer Khatuna Lorig.

From this article:

To prepare for her role as Hunger Games heroine Katniss Everdeen, Lawrence trained with professional archer and Olympian Khatuna Lorig to learn how to shoot a bow and arrow properly.

Edit: The thing with the plane

Since it was mentioned in the comments, I'll say something about the plane.

First of all, a woman is absolutely capable of pulling a 65 pound, since the power to draw such comes from the muscles of the back.

How far can you shoot a 65-pound bow? That depends of course on the arrow, the bow and the draw length of the individual archer. All of these factors are included in the final value: the speed of the arrow. To shorten this, let's take a speed of (feet per second) or. It has the greatest reach when it holds the arc at 45 °:

Okay, there is drag and we have about 400m: if the plane had flown below 400m, she could have shot it.

Also, the speed of the plane shouldn't be that big of an issue when watching videos like this one:

https://www.youtube.com/watch?v=_CyB_L7OoVI

You can hit the exact point where two pendulums overlap. It is normal that then it is also possible to find the right moment to hit something big like an airplane.

starting at 2

Suppose the plane is traveling at 200 mph (I ask a former pilot-neighbor how fast and how low the plane could go on a strafing flight). 200 mph seems like a lower limit. In the scene where Katniss shoots down the plane, it was low but above her. Speaking at 50 feet? I have trouble visualizing how far away it was - say 50 feet? As for a 65 pound bow, is that plausible to a woman, even a woman in great shape? How high / far would the plane have to be for it to be difficult to hit amidships? As for the damage it could do if, as you say, it hits another question.

OddDeer

@ ab2 I answered you in the answer (see under "Edit:") :)