I recently came across the following relationship: let $\Sigma_g$ denote the solid surface of genus $g$. For example: $\Sigma_0$ is the closed ball and $\Sigma_1$ is the solid torus. Then $\partial \Sigma_g$ is the surface of genus $g$ and,
$$ H_k(\partial \Sigma_g)\cong \begin{cases} \mathbb{Z} & k=0\\ \mathbb{Z}^{2g} & k=1\\ \mathbb{Z} & k=2 \end{cases} $$ Since $\Sigma_g$ is homotopy equivalent to the wedge sum of $g$ circles, we also have:
$$ H_k(\Sigma_g)\cong \begin{cases} \mathbb{Z} & k=0\\ \mathbb{Z}^{g} & k=1\\ 0 & k=2 \end{cases} $$ Hence:
$$ H_{k}(\Sigma_g) \oplus H_{2-k}(X)\cong H_k(\partial \Sigma_g) $$
The relationship also holds for a closed disc and its boundary $S^1$. My question is therefore this: is the following true in any generality or is there perhaps a type of space it holds for?
$$ H_{k}(X) \oplus H_{n-k}(X)\cong H_k(\partial X) $$
For lots of interesting spaces, the boundary is empty, but the spaces nonetheless can have quite intriguing homology groups. So your conjecture isn't true for any "interesting" class of spaces that jumps to (my) mind.
In particular, if $X$ is a space with boundary $\partial X$, then there's a space (called the "double" of $X$) consisting of two copies of $X$ glued together along their common boundary. The double of $X$ has (VERY roughly) about twice the homology of $X$, but it has no boundary at all.
For relationships between the homology of a space and its boundary, Lefschetz duality (https://en.wikipedia.org/wiki/Lefschetz_duality) is probably the thing you care about, although Alexander Duality may also prove interesting to you, as you're discussing shapes that happen to be nicely embedded in Euclidean space.
I want to add that I think it's great that you're noticing patterns like this --- it's part of learning to do mathematics. But you probably want to practice a bit more the business of checking a few examples before jumping to a conjecture. One example might be fun; two is a interesting coincidence. Twenty examples make you start looking for a theorem. :)