Two random variables and their sum

Yes. For instance, let $X$ be uniformly distributed on $[0,1]$, and let $Y$ be equal to $\left\{ X+\frac{1}{2}\right\}$, where $\{x\}$ represents the fractional part of $x$. Then $Z=X+\left\{ X+\frac{1}{2}\right\}$ is uniformly distributed on $[1/2,3/2]$. There is an infinite family of such examples, parametrized by the number of jump discontinuities of $Y(X)$, in which $Z$ is uniformly distributed over $[1-1/n,1+1/n]$ and the correlation between $X$ and $Y$ approaches $-1$ as $n\rightarrow\infty$.

We produce a discrete example, because the details are more pleasant to verify. The same idea leads to a continuous example.

Let $X=-3,-1,1,3$ each with probability $1/4$. Suppose that when $X=-3,-1,1,3$ respectively, then $Y=1,3,-3,-1$ respectively. Then $X+Y$ takes on the values $-2$ and $2$, each with probability $1/2$.

Since $X$ and $Y$ have mean $0$, their covariance is $E(XY)$ is $-3$. The variance of $X$ and of $Y$ are each $5$. So the correlation coefficient of $X$ and $Y$ is $-\frac{3}{5}$.

Continuous examples can be made using a similar strategy.