In a recent paper A set of 12 numbers is not determined by its set of 4-sums (in Russian), Isomurodov and Kokhas identify an error in the argument of Ewell (1968) and present two distinct sets of 12 integers with the same multiset of 4-sums.
They also give a nice overview of the area and provide useful references. For the general case $(n,k)$, they mention the following results:
- Selfridge and Straus (1958) describe a union of Diophantine equations such that for $n,k$ not satisfying any of the equations (a typical situation), the set $X$ is unique. For $n,k$ satisfying at least one of the equations, the uniqueness of $X$ remains an open question (the case $(n,k)=(12,4)$ was such).
- Fraenkel, Gordon, and Straus (1962) prove that for each $k$, there exists only a finite number of values of $n$ such that $X$ is not unique.