Summary: Let \(\Gamma =(V,E)\) be a graph of order \(n\). A distance magic labeling of \(\Gamma\) is a bijection \(\ell \colon V \to \{1,2, \ldots, n\}\) for which there exists a positive integer \(k\) such that \(\sum_{x \in N(u)} \ell(x) = k\) for all vertices \(u \in V\), where \(N(u)\) is the neighborhood of \(u\). A graph is said to be distance magic if it admits a distance magic labeling. The Hamming graph \(\text{H}(D,q)\), where \(D, q\) are positive integers, is the graph whose vertex set consists of all words of length \(D\) over an alphabet of size \(q\) in which two vertices are adjacent whenever the corresponding words differ in precisely one position. The well-known hypercubes are precisely the Hamming graphs with \(q = 2\). Distance magic hypercubes were classified in two papers [\textit{Š. Miklavič} and \textit{P. Šparl}, ibid. 44, No. 1, 17--33 (2024; Zbl 08038074); \textit{S. Cichacz} et al., ibid. 36, No. 2, 299--308 (2016; Zbl 1338.05226)]. In this paper we consider all Hamming graphs. We provide a sufficient condition for a Hamming graph to be distance magic and as a corollary provide an infinite number of pairs \((D, q)\) for which the corresponding Hamming graph \(\text{H}(D,q)\) is distance magic. A folded hypercube is a graph obtained from a hypercube by identifying pairs of vertices at maximal distance. We classify distance magic folded hypercubes by showing that the dimension-\(D\) folded hypercube is distance magic if and only if \(D\) is divisible by 4.