Summary: In this paper we investigate bounded additivity in Discrete Tomography. This notion has been previously introduced in [the authors, Lect. Notes Comput. Sci. 7749, 288–299 (2013; Zbl 1339.68256)], as a generalization of the original one in [P. C. Fishburn et al., SIAM J. Appl. Math. 50, No. 1, 288–306 (1990; Zbl 0695.44003)], which was given in terms of ridge functions. We exploit results from [the authors, “Explicit determination of bounded non-additive sets of uniqueness for four X-rays”, in: Proceedings of the 8th international IEEE symposium on image and signal processing and analysis, ISPA’13. Los Alamitos, CA: IEEE Computer Society. 588–593 (2013; doi:10.1109/ISPA.2013.6703808); Discrete Appl. Math. 183, 20–30 (2015; Zbl 1319.68233); Lect. Notes Comput. Sci. 8668, 226–237 (2014; Zbl 1339.68257)] to deal with bounded \(S\) non-additive sets of uniqueness, where \(S \subset \mathbb{Z}^n\) contains \(d\) coordinate directions \(\{e_1, \ldots, e_d \}\), \(| S | = d + 1\), and \(n \geq d \geq 3\). We prove that, when the union of two special subsets of \(\{e_1, \ldots, e_d \}\) has cardinality \(k = n\), then bounded \(S\) non-additive sets of uniqueness are confined in a grid \(\mathcal{A}\) having a suitable fixed size in each coordinate direction \(e_i\), whereas, if \(k < n\), the grid \(\mathcal{A}\) can be arbitrarily large in each coordinate direction \(e_i\), where \(i > k\). The subclass of pure bounded \(S\) non-additive sets plays a special role. We also compute explicitly the proportion of bounded \(S\) non-additive sets of uniqueness w.r.t. those additive, as well as w.r.t. the \(S\)-unique sets. This confirms a conjecture proposed by P. C. Fishburn and L. A. Shepp [in: Discrete tomography. Foundations, algorithms, and applications. Basel: Birkhäuser. 35–58 (1999; Zbl 1105.92332)] for the class of bounded sets.