The best result that I know is that when $X$ simply-connected manifold with non-positive sectional curvature ($n \geq 2$), $q_X = \sqrt{3}$. This directly follows from a corollary of the Rauch comparison theorem. For a reference, see Corollary 2.1 of these lecture notes 1.

To explain why this is the best that I was able to do, let me construct two examples where $q_X \approx 2$, and explain what I need to assume to rule these out. There seems to be two obstacles to bounding $q_X$. One of these is controlled by the global topology and the other is controlled by the more local geometry.

For an example in which $q_X$ can be 2 by the global topology, consider a graph on 4 vertices (say $(v_1, v_2,v_3,v_4)$ which is complete except for a single edge missing (say between $v_2$ and $v_4$). Put the standard metric on this graph assuming all the edges have length 1.
Then the intersection of the balls of radius 1 centered on $v_1$ and $v_3$ is the vertices $v_2$ and $v_4$, as well as the entire edge connecting $v_1$ and $v_3$. This set has diameter 2, because the distance from $v_2$ to $v_4$ is 2. For a picture of such a graph, see the edit below. It's also possible to thicken up this graph and make it into a smooth manifold. Here, the space is not simply connected, and that's allowing us to take alternate paths to points. There is no homotopy between these paths giving "in-between" paths, so $q_X$ is large.

However, there is another obstruction. Even if the space is simply connected and smooth, we may still have $q_X > 2 - \epsilon$. For an example of this, consider the surface given by the graph of
$$ f(x,y) = 20 e^{-((x+2)^2+y^2)} + 20 e^{-((x-2)^2+y^2)}.$$
with the metric from embedding it in $\mathbb{R}^3$. For an intuitive picture, this looks like a mostly flat world with two towering mountains on it. If I consider the points $p =(0,-2)$ and $ q =(0,2)$, then the intersection of the balls of radius just slightly larger than $\sqrt{29}$ contains the points $(-5,0)$ and $(5,0)$. However, in this world, the shortest path from $(-5,0)$ to $(5,0)$ should go around the mountain, so is close to $2\sqrt{29}$ in length. What's going on is that people living at $p$ can use the valley between the mountains to travel to $q$, so the two points are relatively close. In many ways, this closely resembles the earlier example with the graph. It is now possible to what were formerly the loops on the graph, but it is now very costly to do so.

These two examples demonstrate that we can only hope to prove upper bound on $q_X$ when $X$ is simply-connected (or perhaps $1$-connected). Otherwise there will probably be counterexamples no matter how nice the geometry looks. To rule out our mountain example, the issue is fundamentally the curvature of the space. It turns out that if the curvature is positive, one can always create examples that resemble this example, so this suggests that we need to consider non-positive curvature.

If we assume non-positive curvature, we can use triangle comparison theorems to rule out such examples. In particular, on a simply-connected manifold of non-positive sectional curvature, the Rauch comparison theorem implies that $q_X \leq \sqrt{3}$. Using small triangles, we can get arbitrarily close to $\sqrt{3}$, so this implies that $q_X = \sqrt{3}$. In strictly negatively curved space, as $d(x,y)$ grows, one gets an upper bound on $q_{x,y}$ converging to 1.

There may be more that can be said. In all the examples I found with positive curvature that had $q_X$ large, I needed the diameter of the space to be sufficiently large so as to fit two mountains. Perhaps if the curvature is allowed to be positive (but bounded), and the diameter is sufficiently small, one can again obtain an upper bound on $q_X$. I'm not sure how to do this though.

$\textbf{Edit:}$

I wanted to include some pictures to help with the intuition. The first is a picture of the graph for the first example. This graphic was provided by Matt F., so all credit for producing it goes to him.

$${\Huge \nabla \! \! \! \! \! \! \! \Delta }$$

If one takes a tubular neighborhood around this graph (as if were made from PVC piping), then it's possible to construct a smooth genus 2 surface with $q_X > 2 - \epsilon$.

For the second example, here is a plot of the function, which shows the two mountains and the valley between them.

If you would rather manipulate the picture yourself, you can run the following line in Mathematica or on Wolfram cloud.

Plot3D[ 20*Exp[-((x - 2)^2 + y^2)] +
20*Exp[-((x + 2)^2 + y^2)], {x, -10, 10}, {y, -10, 10},
PlotRange -> All]