Wait, I think I might get it. How about a psychic is called in to solve a missing persons case, he consults the tarot cards, and tells the cops that Timmy fell into the Old Well. Lo and behold, there he is!
Do I win?
You win if you’d endorse the claim that the psychic actually knew Timmy’s location.
(The example does seem pretty good to me, at any rate; at least, as long as the psychic has a genuine belief in his methods, rather than being a lucky charlatan)
Even if the Psychic knows himself to be a charlatan, it even seems plausible to me to think that the Police know where Timmy is as a result of the Psychic’s testimony.
But most people (philosophers anyway) will not buy this intuition, I suspect.
-FrL-
When Galileo dropped objects of various weights off the Leaning Tower of Pisa and concluded that all objects fall at the same speed irrelevant of their mass, he could not have been correct. Objects only behave this way in a vacuum and Galileo dropped the objects in air. He must have erred in his observations; nevertheless his theory turned out to be true. My college Physics professor called this “The Miracle of Pisa.”
Is this an example of what you’re looking for? I am still a little confused by the question.
There are two ways to put what I’m looking for:
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Cases in which someone knows X, even though they ought not to believe X
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Cases in which P says “Q knows X” even though P believes Q ought not to believe X.
So if Galileo knew that all objects fall at the same rate, then the example you gave is the kind of thing I’m looking for, as per formulation 1. Also, if you yourself think Galileo knew that all objects fall at the same rate, and if you say so, then you’ve provided an example of what I’m looking for, as per formulation 2.
Also, your example is a fascinating one, no matter whether it fits what I’m looking for or not. 
-FRL-
Why?
I don’t know if I understand the question, but it triggered an example that I heard recently, so I’ll tell it.
It involved a nurse working with a critically ill patient. She “knew” (based on medical facts) that this patient was unlikely to live but said that she had a sense of peace (ostensibly from God) that the patient would live. To date, the patient was still alive at the telling of the story.
So the nurse “knew” (and how do you distinguish this from believe) that the patient would live and it turned out to be true (at least for the moment) but the “knowledge” was not justified based on the medical facts of the case.
I suspect for much the same reason you put “know” in quotes in every instance in your post. In other words, you felt the need to mark that you’re not using the word “know” quite properly. In other words, you think you’re not using the word to mean what it really means, properly speaking.
Well, that’s why I think you were using the scare quotes, but I’m not sure. Was there some other reason you were using them?
As to how to distinguish knowledge and belief, I’m actually interested in hearing what you might have to say about that. Sometimes, you say “He believes such and such.” Other times, you say “He knows such and such.” What’s the difference?
-FrL-
Er, well actually I was hoping that you were going to shed some light on this because I’m not a philosopher.
But the reason that I placed the word know in quotes was that I don’t believe that she knew because I don’t share her belief. But then again, some things that people say they know, I don’t agree with either. When people say they know that the world was created in 6000 years, I don’t agree. And while I can say I don’t believe that, I can also say I know that not to be true (from my perspective).
I noticed that in the article that you linked, the author talks about knowledge having a “world connection.” But he doesn’t go very far in defining what this “world connection” consists of. Does that mean that only statements based on the physically ascertainable can be considered knowledge? But then the statement that “I know he loves me” or “I know that I’m in love with him” would have no meaning. Is this just an example of “loose talk”? Do paradigms have a “world connection”? Do models of universes we can’t see have a “world connection”?
I don’t know that answer, but I have lots of questions. 
Here’s an example. Some people say “I believe in God.” But I’ve seen people write “I know that God exists.” And when I’ve asked whether they know it like they know that a tree exists outside their window (if one did), they say yes, they know it in that sense. If pressed to know how they know this, many note physical experiences which caused them to know, so this might be the “world connection” that the author talks about. So does a strong enough belief lead to using the word knowledge? If the personal experiences strung together wouldn’t convince someone else of the same thing, does this still constitute justified belief leading to knowledge?
Do parallel universes exist? How can one know? Do we have to experience them to know? If we devise an experiment that hints at them, is this sufficient?
And how about the fact that we can only experience things through our senses? If these are fallible, can we know anything?
Regarding “connection to the world”: I think the idea is supposed to be that someone who knows thereby has some reliable access to the facts. So, for example, if I know my cat is sitting next to me, it’s because I have reliable access to the facts concerning the position of my cat. If I know Mars has almost no oxygen in its atmosphere, it’s because I’ve got some kind of reliable access to the facts concerning the level of oxygen on Mars. In the case of the cat, this “reliable access” consists in the fact that I have an unhindered visual line between myself and the cat. In the case of mars, this “reliable access” consists in my trusting what I read in certain books.
Meanwhile, if I believe my cat is in my bedroom because I heard a noise coming from in there, then whether my cat really is in there my belief arguably doesn’t consist in knowledge if the source of the noise was something other than the cat. If the noise was from a skunk crawling through the window instead, then my hearing the noise does not constitute any kind of “reliable access” to the facts concerning my cat’s position. My belief “My cat is in my bedroom” is not connected (in the right way) to the actual cat I’ve formed a belief about.
What about “I know I love him?” If there really is a such thing in the world as love, then it is presumeably possible to have reliable access to the facts about whether one loves someone or not. By the way, “world” here doesn’t have to be taken as specifying the physical world.
What about models of universes we can’t see? It depends on the methods used and reasons for arriving at these models. Perhaps they are the inexorable outcome of some math required by physical science? Then I’d say you have the same kind of “reliable access” to the facts about these universes that you have to facts about mathematical entities like numbers and operations and so on. Or perhaps its better to say you have the same “reliable access” to these universes as you have toward facts about gravity on other planets–you can calculate what the gravity must be even if you’ve never actually gone to the planet, dropped things, taken measurements, and so on. Either way, this “reliable access” probably constitutes the kind of “connection to the world” referenced in the paper, and so allows for knowledge rather than mere opinion.
Well, philosophers have opinions about this, and most are in agreement on this particular issue. But I’m reticent to say too much about what the philosophers say, because what I’m trying to find out is what the non-philosophers say. I’m afraid I will prejudice the discussion if I say too much.
In the spoiler box, I have typed out the general consensus amongst philosophers.
So here’s what most philosophers say: No amount of the feeling of certainty makes a belief knowledge. You can feel absolutely certain while failing to have knowledge. And most would also say you can have knowledge even when you feel quite uncertain. What makes something knowledge is whether or not it is true, and whether or not it is believed for the right reasons.
That’s a live issue in contemporary epistemology.
Are you worried about this because you think knowledge requires certainty?
-FrL-
Well, in his article “Elusive Knowledge” David Lewis argues that justification is not necessary for knowledge. He writes:
Incidentally, I am not convinced that his examples are real examples of knowledge without justification (especially if you construe justification and evidence broadly, to include “subtle visual cues” and not just overt beliefs), but there you have it.
In my opinion, it’s not dead; people just don’t call it ordinary language philosophy any more. I think arguments for contextualism, with their appeal to the multifarious ways in which ‘to know’ is used, is as clear a case of ordinary language philosophy as you are likely to see.