In a recent paper [A set of 12 numbers is not determined by its set of 4-sums][1] (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][2]. 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.

  [1]: http://mi.mathnet.ru/eng/znsl6308
  [2]: http://www.mathnet.ru/php/getRefFromDB.phtml?jrnid=znsl&paperid=6308&output=htm&option_lang=eng