I start with some background, but people familiar with the subject may jump directly to the question.
Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an almost hypercomplex structure on $M$ is a 3-dimensional sub-bundle $Q\subset End(TM)$ spanned by three endomorphisms $I$, $J$ and $K$ satisfying the quaternionic identities: $I^2=J^2=-Id$, $IJ=-JI=K$.
An almost quaternionic structure on $M$ is a 3-dimensional sub-bundle $Q\subset End(TM)$ which is locally spanned by three endomorphisms with the above property.
In both cases one may assume (by an averaging procedure) that $M$ is endowed with a Riemannian metric $g$ compatible with $Q$ in the sense that $Q\subset End^-(TM)$, i.e. $I$, $J$ and $K$ are almost Hermitian. Using this one sees that an almost hypercomplex or quaternionic structure corresponds to a reduction of the structure group of $M$ to $\mathrm{Sp}(n)$ or $\mathrm{Sp}(1)\mathrm{Sp}(n)$ respectively, but this is not relevant for the question below.
Notice that in dimension $4$ every manifold has an almost quaternionic structure (since $\mathrm{Sp}(1)\mathrm{Sp}(1)=\mathrm{SO}(4)$), but there are well-known obstructions to the existence of almost hypercomplex structures. For example $S^4$ is not even almost complex. Finally, here comes the question:
Are there any known topological obstructions to the existence of almost quaternionic structures on compact manifolds of dimension $4n$ for $n\ge 2$?
EDIT: Thomas Kragh has shown in his answer that there is no almost quaternionic structure on the sphere $S^{4n}$ for $n\ge 2$. I have found further obstructions in the litterature and summarized them in my answer below.
Best Answer
I know this is a bit late, but as you mentioned Cadek's and Vanzura's paper, I'd like to point out (selfishly?) that there's also my paper which uses a bit of their work and gives some integrality conditions on the existence of quaternionic structures on closed manifolds -- an example is stated below. I should emphasize that I really mean honest quaternionic not just almost quaternionic here, although the referee believed that the same should hold for only almost quaternionic structures too.
Theorem: Let M be an 8-dimensional compact quaternionic manifold with Pontryagin classes $p_1(TM)$ and $p_2(TM)$ and a fundamental class $[M]$. Then the following expressions are integers $$\biggl(\frac{143}{960}p_{1}^{2}-\frac{89}{240}p_{2}\biggr)[M], \quad \biggl( -\frac{17}{480}p_{1}^{2}+\frac{71}{120}p_{2}\biggr)[M].$$