In my philosophy class, there was something called the Heap Problem:
"One grain of sand is not a heap. Adding a grain of sand to something that is not a heap will not make it a heap. By induction, then, heaps cannot exist."
The conclusion, of course, is obviously false, because heaps of sand do exist. But you can't state at what point something that's not-a-heap becomes a heap.
AFAIK, this was still an open question when we covered it in class (2001). I don't know of anyone that's provided a convincing argument for why two premises that seem true result in a conclusion that's obviously false.
Reminds me of Zeno's paradox about Achilles and the tortoise:
Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, for example 10 feet. It will then take Achilles some further time to run that distance, in which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been--he can never overtake the tortoise.
It's because we tend to think in terms of gestalts, and those don't play well with reductionist approaches. It's only a contradiction if you assume that we apprehend all the constituent pieces of the world at once, rather than the abstract whole.
Treating a pile of sand as an accumulation of individual grains which can be precisely abstracted through induction is an unnatural mode of thinking. Without focusing our attention on it, it will remain a single, fuzzy abstraction. That we see no clear distinction between these two modes seems more a neurological phenomenon than a philosophical one.
An interesting instance of this arose when determining how numbers are described in the Piraha language, a language with only three words for quantities. Seeing one battery, the Piraha called it "ho'i". When they added one more (a large increase percentage-wise), they immediately switched to another word, so therefore "ho'i" means "one." But when they started with ten batteries and started removing them, one of them started calling it "ho'i" at six batteries, so therefore "ho'i" really means "few." (http://en.wikipedia.org/wiki/Pirah%C3%A3_language#Numerals_a...) I'd explain that as: when deciding whether to call something "few" or "many," they anchored off the initial judgement, and looked at the percentage change. So one politician could convince voters a 10% tax increase will have no effect by starting off "Well, what would a 0% increase do? How about 1%?", while another could convince them it would be devestating by starting off at 20%.