Are there statistical paradoxes where "no correlation" is shown, when there really is one?

I don't know much about statistics. I heard that the Simpson's Paradox makes it look like the correlation is reversed when more than one group is combined, negative instead of positive or vice versa. Is there a statistical paradox where there is no correlation shown at all, when there really is one? Or a statistical paradox when you fail to reject the null, even when your statistical power is very good?