In this article, we study the cardinality of the intersection of multiple q-ary Hamming balls for q ≥ 3. The problem has previously been studied in the binary case and for two balls in the case of q ≥ 3. When each ball has radius t and they are centered at words of a set S, we present a link between the asymptotic size of the cardinality and the center of the set S. For exactly three balls, we consider the largest and smallest possible intersection sizes and possible sets S leading to them. The intersections of Hamming balls have been the focus of multiple studies recently, due to their connections to Levenshtein’s sequence reconstruction problem and DNA memory systems, where the information is stored into DNA strands. The case with q = 4 is especially important for applications related to DNA due to the four nucleotides of DNA.