Does $2 \mathrm{Cov}(X, Y) \preceq \mathrm{Cov}(X, X) + \mathrm{Cov}(Y, Y)$ hold for random vectors?

Question

For any two $n$-dimensional random vectors $X$ and $Y$, does the following inequality hold?

$$2 \mathrm{Cov}(X, Y) \preceq \mathrm{Cov}(X, X) + \mathrm{Cov}(Y, Y),$$

where $A \preceq B$ means $A - B$ is positive semi-definite.

Thanks!