Let $X, Y$ be two real-valued random variables. We have that $$COV(X,Y)\le \sqrt{\text{Var}[X]\text{Var}[Y]}\le\max\{\text{Var}[X],\text{Var}[Y]\}\le\text{Var}[X]+\text{Var}[Y].$$
This allows us to write: $$ \text{Var}[X+Y] = \text{Var}[X]+\text{Var}[Y] + 2COV(X,Y)\le 3(\text{Var}[X]+\text{Var}[Y]). $$
This seems like a very crude bound. Is there a way to improve it? Since I'm only interested in bounding it as a function of $\text{Var}[X]+\text{Var}[Y]$, I'll phrase the question as:
What is the smallest constant $c$ such that $\text{Var}[X+Y]\le c\cdot (\text{Var}[X]+\text{Var}[Y])$ for any two random variables $X,Y$?
By setting $X=Y$, we have $\text{Var}[X+Y]=\text{Var}[2X]=4\text{Var}[X]=2(\text{Var}[X]+\text{Var}[Y])$ so we have $2\le c\le 3$. What is the correct answer?
$c=2$ works. (as in the case $(a+b)^2 \leq 2(a^2+b^2)$ of squares.)
Note that, assuming without loss of generality for the proof that $\mathbb{E}[X]= \mathbb{E}[Y]=0$, $$\begin{align} \operatorname{Var}[X+Y] &= \mathbb{E}[(X+Y)^2]\\ &= \mathbb{E}[X^2]+\mathbb{E}[Y^2]+2\mathbb{E}[XY] \\ &\leq \mathbb{E}[X^2]+\mathbb{E}[Y^2]+\left(\mathbb{E}[X^2]+\mathbb{E}[Y^2] \right)\tag{AM-GM}\\ &= 2\left(\mathbb{E}[X^2]+\mathbb{E}[Y^2]\right)\\ &= 2\left(\operatorname{Var}[X]+\operatorname{Var}[Y]\right) \end{align}$$ where te AM-GM inequality was used to write $$ \mathbb{E}[XY]\leq \mathbb{E}\!\left[\frac{X^2+Y^2}{2}\right]\,. $$