Next time someone says "But even the ancient Greeks knew the Earth was round!" I'm going to link them to this thread.
I'm not even sure if that is correct. They certainly believed the Earth to be round, and their belief turned out to be correct, but some of the reasons for their belief were simply silly. I think their main reason was that they believed the sphere to be the most perfect form in nature; certainly that was one of their main reasons.
For a correct belief to actually constitute knowledge, you have to know why the belief is true, and you have to be right about that.
For example, I may believe I'm never going to win the lottery. Since I will never win the lottery, this is a correct belief, but if my reasoning is that the stars are against me, cursing me with bad luck, then this is simply a silly belief of mine. On the other hand, if my reasoning is that I know I will never buy a lottery ticket because the expected return is much lower than the price, then it is indeed knowledge.
So one can never know something which is actually false, although one may think one does.