They do. Simply choose, say, $X_n$ a Bernoulli random variable taking values $n$ or $-n$ with probability $\frac 12$ for each. Notice that $Var(X_n) = n^2$. Define $Y_n = - X_n$. You have $Var(X_n + Y_n) = 0$ and, to answer your last interrogation, $Cov(X_n,Y_n) = - Var(X_n) = -n^2$.
Hachino
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