Humans can become dizzy by spinning around quickly. Small children will occasionally do this on purpose, for the thrill.
However, intentional dizziness inducing behavior is (as far as I am aware) never observed in cats or kittens. (Although I did see a particularly spectacular quadruple spin after one of my kittens collided with another in midair.) It would also be harder to observe the effects of dizziness as cats are quadrupedal and so are more stable than bipedal humans.
According to some not-always-reliable sources on the internet (such as Yahoo Answers or YouTube), cats can become dizzy, but recover from it much more rapidly than humans; but these accounts appear to be merely anecdotal, with others reporting the opposite, that despite their attempts they have been unable to induce observable dizziness in their cats.
It is a known fact that cats can become dizzy due to things like ear infections or medication side-effects, and that motion can aggravate it (link), so it would seem plausable that a cat could become dizzy by spinning fast enough for long enough even in absence of those factors. However, I can't find any non-sketchy sites that give a clear answer about this.
Can cats become dizzy due to spinning, in absence of disease or other special circumstance? How quickly do they recover from dizziness compared to humans?