Is it possible that X and Y are uncorrelated but X can significantly predict Y? I was just wondering is it possible that X and Y are uncorrelated but X can significantly predict Y?
If so, how would you explain and interpret that?
 A: Yes.
Take for example the unit circle coordinates for (x, y).  They are uncorrelated, yet if you know x you know that y can only take on 2 values -  or 1 exactly (y=0) if x =1 or -1.
More generally marginal independence does not imply conditional indedepedence. That is there might be a third variable z that allows you to predict y from x knowing z.
See Examples of marginal independence, conditional dependence
A: Let's prove the fact that @whuber gave in a comment to the question:

Let $X$ be symmetrically distributed around $0$ and let $Y=X^2$. The
latter is perfectly predictable from $X$ but the correlation is zero.

We first note that $X$ obviously determines its square $X^2=Y$. The proof of the second part is more interesting:
By definition, a random variable $X$ has a symmetric distribution about $\mu$ if $X-\mu$ has the same distribution as $\mu-X$. For symmetry about $\mu=0$ we get that $X-0=X$ has the same distribution as $0-X=-X$.
Using that $X$ and $-X$ have the same distribution and thus the same moments, the fact that $(-1)^a=-1$ for any odd number $a$, and linearity of expectation, we get
$$\mathbb{E}(X^a)=\mathbb{E}[(-X)^a]=\mathbb{E}(-X^a)=-\mathbb{E}(X^a)$$ and hence $\mathbb{E}(X^a)=0$ for any odd number $a$.
For the covariance between $X$ and $Y$ we have
$$\mathrm{Cov}(X,Y)=\mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y)=\mathbb{E}(X^3)-\mathbb{E}(X^1)\mathbb{E}(X^2)$$
and using the previous result $\mathbb{E}(X^a)=0$ for $a\in\{1,3\}$ it follows that $\mathrm{Cov}(X,Y)=0$, which is equivalent to $\mathrm{Corr}(X,Y)=0$.
This is not particularly surprising as the covariance and correlation are measures of linear association between two random variables.
A: Correlation measures a certain kind of relation (usually linear) between variables, but other relationships (nonlinear) are possible as well. This is discussed in depth in the Why zero correlation does not necessarily imply independence thread. So it is possible that there’s a non-linear relationship between the two variables, that makes them non-independent, yet it is not measured by correlation.
