Let $(X_n)$ and $(Y_n)$ be two sequences of random variables defined on the same probability space such that the variance of all components $X_n$, $Y_n$ is finite and the sequence of variances of $X_n+Y_n$ is non-zero and bounded. Does this imply that the variance sequences of $X_n$ and $Y_n$ are bounded as well?

In formulas, let $a_n^2$ and $b_n^2$ be the variance of $X_n$ and $Y_n$, respectively, and let $c_n$ be the covariance of $X_n$ and $Y_n$. Then the assumption from above translates into

$$0 < \operatorname{Var} (X_n+Y_n) = a_n^2 + b_n^2 + 2c_n \leq M$$

for some $M \geq 0$. From a purely analytical point of view, one could take $a_n^2=b_n^2=n+M/2$ and $c_n=-n$. Then $(X_n)$ and $(Y_n)$ would have unbounded variances but the variance of $(X_n+Y_n)$ would be bounded. Do random variables of this form exist?