Example: if I find a statistically significant difference between the heights of men and women, does this say something about being able to predict whether a person is a man or woman based on the height?
It seems to me that in every case where there is a statistically significant difference, we should be able to make a prediction (classify something, and evaluate it using cross-validation) of the independent variable based on knowing the dependent variable. I couldn't think of a counterexample after a long period of thinking about it.
Another example: there is a linear correlation between $A$ and $B$. Given $B$, I would be able to predict $A$ with high certainty.