If an ngon solution can’t be found, shouldn’t the tool automagically tesselate the problematic face instead?
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Can you show an example of what you are talking about?
So I finally managed to get a case where it fails again, but also discovered that in this instance it might not be a “real” case:
Seems everything here was separated, and the knife tool snapped to the vertices of two different faces instead of the middle one I dragged across. Joining vertices by distance solved things in this case, but still, the problem isn’t immediately apparent to the user, because it still appears as if the tool fails.
It still would be much neater if the cut happened even in the above example.
EDIT: What the heck… you’re not allowed to upload either .blend or .zip to this forum?
Not sure if it’s that what you mean. But the knifetool sometimes adds all vertices of a cut without adding all edges of the cut.
It’s quite easy to reproduce by trying the shape I’ve drawn here a couple of times on a defaultcubes face.
Not happening all the time though
To upload a blender file use “pasteall.org”, it has a .blend option.
Regarding the example, have you checked if that upper vertice is connected to the edge it seems to be connected, that behaviour happens when you try to cut over elements that are not connected, so moove the upper vertice and check if it’s connected of if there some empty space 
Fascinating. Seems neither of you guys read/understood what I actually posted.
Anyway, thanks for the pasteall tip, I’ll upload a file when I find a better example.
We did read what you wrote, but your info is too limited to reproduce it.
You are just telling us that an ngon would be created by the cut.
What kind of face gets cut in your scenario, also an ngon, a quad, a tri? Are you talking of concave or convex cases of the Ngon? Is the the Ngon there before the cut or part of a complex cut, or a result? Is the cut a a closed cycle? Does it start/end on existing edges or verts? Is the Ngon a flat polygon or not? Was your mesh perhaps also disconnected as in your example? Is the ngon the inside your closed cut or on the outside of a closed cut, is the cut crossing itself,…
I posted my case because its a scenario where at least parts of the cut don’t get cut. So I though it might relate to what you are experiencing. If you have a more specific case in mind you should try to explain or really post a video of it. Splitting Ngons by itself works well here aswell as getting ngons as result.
If you think we did not understood the problem why don’t you try to explain it in a different way?
From my user perspective, I don’t see a difference here and I don’t think there should be a difference:
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Because what you are missing there is that the knife is snapping to the vertice of one of the left polygons and it’s wanting to cut into the “parent” polygon or a connected one, not an external polygon from a different island, even if it’s in the same object, but if you enable the “Cut Through” option with the Z key, you will get the cut you want
But be aware that you will be cutting also other polygons behind that one.
Yes it fails here. You played off one feature against the other, so that it fails in this case. You force snapping to unconnected verts that hinders the cut line to cut the center face from outside to outside nor does the line start and end on the edge of center face with its start and end point. Just for demonstration purposes, if you move theses outer faces apart from the center one it will work. Didn’t test cut through as Juan suggested, but yes that might be a workaround.
But concerning your inital question I don’t think tesselating the center quad will help here. Do you think this relates to your initial problem? 
My problem is that as a user, I don’t care about the technical details of the underlying topology. Who knows, I might want it that way!
The knife tool clearly snapped to something. I continued to click so I clearly wanted the tool to do something, yet it did nothing.
That is a poor user experience.
(Also, I absolutely don’t want to “cut through”. I just want the tool to cut across the clearly visible polygon that it ignored in my second case.)
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Yes, you stated this before. People try to explain it to you to give you the info needed to circumvent it or at least handle problematic cases better. If you’re not interested, ok. But why do you ask in the threads title for it then?
Sure, we all love tools tools without any flaws. Not that realistic as assumption though.
If you don’t like cut through, then hide the outer faces before you cut it.
It doesn’t look like the hardest case to include. Howard did signalize interest already, so let’s see what he says.
I’m just providing feedback from the perspective of a new user. Blender wants new users, right?
Hopefully, they want to improve too (and as I improve, my feedback will go away).
I updated the wording of the topic title now.
Ok. Hopefully yes. 
I understand what you mean, but I don’t think it’s a failure, as I already explained it cuts only in the connected geometric piece, if you want to cut along other disconnected pieces you have to enable the option I told you.
It’s not a fail, it’s the way it’s designed, like it or not, and new users are always welccome, but new users have to learn how to use tools that already exist in Blender, but if those new users want to adapt the tools to their liking, it’s perfectly fine to ask for that and create a conversation around it.
Just the thing is that the answer you will receive to this thread title is “it’s not failing, it’s working as designed”.
But if you want some positive reaction a much better title would be “Improvement for Knife to allow it to cut along near disconnected surfaces” or something like that, it’s a lot different and much more positive, and in the end you are saying more or less the same
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Using Blender 2.91 and one of the first things I run across is the knife tool’s inability to cut Ngons again…
It’s just one of those basic things that drives you up the wall when you try to use Blender.