The notion of approximate resolution was introduced and investigated in earlier papers of S. Mardešić and the authors [\textit{S. Mardešić} and \textit{T. Watanabe}, Glas. Mat., III. Ser. 24(44), 587--637 (1989; Zbl 0715.54009); \textit{T. Miyata} and \textit{T. Watanabe}, Topology Appl. 113, No. 1--3, 211--241 (2001; Zbl 0986.54033); \textit{T. Miyata} and \textit{T. Watanabe}, Topology Appl. 122, No. 1--2, 353--375 (2002; Zbl 1024.54016)]. In particular the authors investigated two metrics in a compact metric space: 1. \(d_{\mathbb{U}}\) -- induced by a normal sequence \(\mathbb{U}\), 2. \(d_{\mathbf{p}}\) -- induced by an approximate resolution \(\mathbf{p}\). That allowed the authors to define 1. a \((\mathbb{U},\mathbb{V})\)-Lipschitz map, i.e., a Lipschitz map with respect to metrics \(d_{\mathbb{U}}\), \(d_{\mathbb{V}}\) for normal sequences \(\mathbb{U}\), \(\mathbb{V}\), 2. a \((\mathbf{p},\mathbf{q})\)-Lipschitz map, i.e., a Lipschitz map with respect to metrics \(d_{\mathbf{p}}\), \(d_{\mathbf{q}}\) for approximate resolutions \(\mathbf{p}\) and \(\mathbf{q}\). In the paper under review, the authors continue this approach and generalize the notion of box-counting dimension 1. for every compact metric space with a normal sequence, 2. for approximate resolutions of any compact metric space. The authors prove that both introduced dimensions are Lipschitz subinvariant with respect to the respective Lipschitz maps mentioned above. Moreover, they show some fundamental properties of the box-counting dimension of an approximate resolution: the subset theorem, the product theorem and the sum theorem.