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Now I know Gettier presents cases of JTB that are not cases of knowledge (e in case I, h in case II), but the other way round?
For a proposition p not JTB, we need either p is not true or S doesn't believe that p or S is not justified in believing that p.
Can p still be knowledge?
Knowledge is never justified true belief (JTB).
Justification is an alleged process that shows knowledge is true or probably true. Knowledge is never justified because justification is impossible, unnecessary and undesirable. If you assess ideas using argument then the arguments have premises and rules of inference and the result of the argument may not be true (or probably true) if the premises and rules of inference are false. You might try to solve this by coming up with a new argument that proves the premises and rules of inference but then you have the same problem with those premises and rules of inference. You might say that some stuff is indubitably true (or probably true), and you can use that as a foundation. But that just means you have cut off a possible avenue of intellectual progress since the foundation can't be explained in terms of anything deeper. And in any case there is nothing that can fill that role. Sense experience won't work since you can misinterpret information from your sense organs, e.g. - optical illusions. Sense organs also fail to record lots of stuff that does exist, e.g. - neutrinos. Scientific instruments aren't infallible either since you can make mistakes in setting them up, in interpreting information from them and so on.
Knowledge need not be true. Newtonian mechanics was knowledge, but it was never true. Also, quantum mechanics and general relativity contradict one another but we don't have replacements for either of them. they are both false but they are also both knowledge.
Knowledge also need not be belief. There is a lot of knowledge encoded in computer programs or books that nobody believes. The point of writing stuff down is so that you don't have to remember it.
Knowledge consists of solutions to problems, not JTB.
For more criticisms of the JTB theory see "Realism and the Aim of Science" by Karl Popper, "The Beginning of Infinity" by David Deutsch, http://www.curi.us/1232-justified-true-belief-speech.
No, all of the examples in "Is Justified True Belief Knowledge" are examples of the form where the proposition is a JTB, but fail to be (commonly percieved as) cases where the person had knowledge.
One way of interpreting this is that Gettier showed that naive JTB (or anything like it, i.e. the three definitions indicated at the start of the paper) is at best a necessary, but not a sufficient, condition for knowledge.
Let me complete Dave's answer by discussing whether JTB is necessary for knowledge.
Obviously something cannot count as knowledge if it is false.
Some kinds of "know-how" may not count as beliefs, but we're talking about propositional knowledge here and I doubt that one can know something to be true while not believing it (arguably the colloquial sense "I know it but I can't believe it" expresses astonishment rather than a lack of commitment). It seems hard to make sense of knowledge without somehow relating it to belief. A possible case is when you don't have second order knowledge (you don't know that you know). E.g.: you learnt the capital of a foreign country by heart a long time ago and still can tell it automatically, while not being sure it's the right answer. But isn't it a case of "know-how"?
Finally justification of a belief means that it is rational for you to believe it, that you have good reasons to believe it. Maybe these reasons are not accessible to you (as externalists would have it) but it seems doubtful that an irrational belief, something you believe randomly, can count as true knowledge. Some counterexamples might involve mystic or supernatural intuitions, but it's not clear that we would talk about knowledge (instead of strong belief or faith) in such cases.
