Bounds on quantiles of the sum of (possibly dependent) random variables Suppose I have two continuous random variables $X$ and $Y$, and I can evaluate the quantiles of these distributions individually. I aim interested in what kinds of constraints can be put on quantiles of $(X+Y)$ given knowledge of the quantiles of $X$ and $Y$ individually -- without making any assumptions about $X$ and $Y$ being independent, or having some particular dependence structure.
Specifically, denote the $\alpha$ quantile of $X$ as $Q^{\alpha}_{X}$, which satisfies $Pr(X<Q^{\alpha}_{x}) = \alpha$. Similarly, let $Q^{\alpha}_{Y}$ be the $\alpha$ quantile of $Y$. I assume I can evaluate these for any $\alpha$.
My question is, what bounds can we place on $Q^{\alpha}_{X+Y}$ in terms of the quantiles for each distribution individually? For example, I think we can say that (supposing $\alpha > 0.5$):
$$ Q^{(1 - 2(1-\alpha))}_{X+Y} \le Q^{\alpha}_{X} + Q^{\alpha}_{Y}$$
A specific instance of this would be:
$$ Q^{0.95}_{X+Y} \le Q^{0.975}_{X} + Q^{0.975}_{Y}$$
I'll explain why I think that below, but my question is are there stronger constraints I can make on the quantiles of the sum, supposing I know $X$ and $Y$ exactly, but without making any assumptions regarding the dependence of $X$ and $Y$?
The reason I think the above result is true: Suppose I have an (arbitrarily large) sample of $X+Y$, denoted $(x_{1}+y_{1}), (x_{2}+y_{2}), ..., (x_{i}+y_{i}), ...$. Then in the latter sample, a fraction $1 - \alpha$ of the terms should have $x_{i} > Q^{\alpha}_{x}$, and a fraction $1 - \alpha$ of the terms should have $y_{i} > Q^{\alpha}_{y}$. Thus there must be (irrespective of dependence) less than a fraction $2(1-\alpha)$ terms in the latter sample that have one or both of $x_{i} > Q^{\alpha}_{x}, y_{i} > Q^{\alpha}_{y}$. We know that all the other terms have smaller $x_{i},y_{i}$, so there must be at least a fraction $(1 - 2(1-\alpha))$ that is less than  $Q^{\alpha}_{x} + Q^{\alpha}_{y}$
 A: So we have random variables $X, Y$ with unknown dependence structure. Say they
have cumulative distribution functions $F_{X}, F_{Y}$ respectively. We are
interested on bounds that can be placed on quantiles of the random variable $ X+Y$.
Let $Q_{X}^{u}$ be the $u$ quantile of $X$, i.e. $$ F_{X}(Q^{u}_{X}) = Pr(X \le Q^{u}_{X}) = u $$, and similarly let $Q^{v}_{Y}$ be the $v$ quantile of $Y$.
From the theory of copulas, there is a copula $C_{XY}(u,v)$, defined as:
$$C_{XY}(u,v) = Pr(X \le Q^{u}_{X}, Y \le Q^{v}_{Y})$$
The Frechet-Hoeffding bounds for a copula imply that:
$$ \max\Big{(} 1 - (1-u) - (1-v), 0 \Big{)} \le C_{XY}(u,v) \le \min(u,v) $$
, see https://en.wikipedia.org/wiki/Copula_(probability_theory)#Fr.C3.A9chet.E2.80.93Hoeffding_copula_bounds. 
In the above inequality, the upper-bound corresponds to the comonotonic case, mentioned
in some comments. The left-hand-side bound is a generalisation of arguments
like those used to provide bounds in the question (although the latter assumed $u = v$, whereas the Frechet-Hoeffding bounds do not require that).
Focussing on the left-hand-side of the above, define $w = 1 - (1-u) - (1-v)$. Suppose
$w$ is fixed, and so $u,v$ are now restricted to meet this requirement (i.e. $v = 1 - u + w$; $u \in [w, 1]$ ). From the Frechet-Hoeffding bounds, we know that:
$$ \max(w, 0) \le C_{XY}(u, 1 - u + w) = Pr(X \le Q^{u}_{X}, Y \le Q^{1 - u + w}_{Y}) \le \min(u, 1 - u + w) $$
In other words, the probability that both $X \le Q^{u}_{X}, Y\le Q^{1-u+w}_{Y}$ is at least $w$ -- and so crucially, the probability that the sum $X+Y$ is less than the sum of these quantiles must also be bound from below by $w$.
This means that for any such $w, u$, we know for sure that:
$$ Q^{w}_{X+Y} \le Q^{u}_{X} + Q^{1 - u + w}_{Y} $$
because both the $X$ and $Y$ values must individually be less than their
respective quantiles appearing in the above expression.
Since this relationship is true for any $w, u$, if we have a fixed $w$ we can
test all values of $u$ and find the one that minimises the RHS, i.e.
$$ Q^{w}_{X+Y} \le \min_{u \in [w,1]} \Big({} Q^{u}_{X} + Q^{1 - u + w}_{Y} \Big{)} $$
This is a stronger bound that that presented in the question, which effectively
assumed $u = v$ (or equivalently, u = 1 -u + w).
