Sometimes you want prove something or solve a problem, but at last you find out it’s just a paradox. Just like the case when you solve the equation $x^2 + 1 = 0$. People without knowledge of complex numbers surely will regard it as impossible, and for people who know imaginary numbers, they still have no idea about how big x is, even they know $x = i$ or $-i$.
Another case is a confidence interval [0, 1] for a proportion statistic p. It doesn’t violate any statistical rules, but it’s of no use.