I would be happy about an answer addressing general dimensions, but I am mostly interested in $d=4$, so I will state the question for this case.
Question: Given two closed sets $A,B\subset\Bbb R^4$ so that $A$, $B$ and $A\cap B$ are homeomorphic to 4-balls, then is $A\cup B$ also homeomorphic to a 4-ball?
If it helps, and to avoid pathologies, let's assume that everything is PL, i.e. all these sets are PL-homeomorphic to PL-balls.
Some thoughts
Seifert-van-Kampen suggests that the union has trivial fundamental group. That is a good start.
I though about first proving that $B\setminus A\simeq D^4$ (the 4-ball) or that $\partial A\cap\partial B\simeq S^2$ (the 2-sphere), but both are not necessarily true, already in small dimensions.
