The problem of reconstructing finite subsets of the integer lattice from X-rays has been studied in discrete mathematics and applied in several fields like data security, electron microscopy, and medical imaging. In this paper, we focus on the stability of the reconstruction problem for some special lattice sets. First we prove that if the sets are additive, then a stability result holds for very small errors. Then, we study the stability of reconstructing convex sets from both an experimental and a theoretical point of view. Numerical experiments are conducted by using linear programming and they support the conjecture that convex sets are additive with respect to a set of suitable directions. Consequently, the reconstruction problem is stable. The theoretical investigation provides a stability result for convex lattice sets. This result permits to address the problem proposed by Hammer (in: Convexity, vol. VII, Proceedings of the Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, 1963, pp. 498–499).