Aminoglycosides (AGs) in combination with β-lactams play an important role in antimicrobial therapy in severe infections. Pan-resistance to clinically relevant AGs increasingly arises from the production of 16S rRNA methylases (RMTases) that are mostly encoded by plasmids in Gram-negative bacteria. The recent emergence and spread of isolates encoding RMTases is worrisome, considering that they often co-produce extended-spectrum β-lactamases (ESBLs) or carbapenemases. Our study aimed to retrospectively analyze and characterize the association of carbapenem- and aminoglycoside-resistant clinical isolates in Switzerland during a 3.5-year period between January 2017 and June 2020. A total of 103 pan-aminoglycoside- and carbapenem-resistant clinical isolates were recovered at the NARA (Swiss National Reference Center for Emerging Antibiotic Resistance) during the 2017–2020 period. Carbapenemase and RMTase determinants were identified by PCR and sequencing. The characterization of plasmids bearing resistance determinants was performed by a mating-out assay followed by PCR-based replicon typing (PBRT). Clonality of the isolates was investigated by multilocus sequence typing (MLST). Over the 991 Enterobacterales collected at the NARA during this period, 103 (10.4%) of them were resistant to both carbapenems and all aminoglycosides. Among these 103 isolates, 35 isolates produced NDM-like carbapenemases, followed by OXA-48-like (n = 23), KPC-like (n = 21), or no carbapenemase (n = 13), OXA-48-like and NDM-like co-production (n = 7), and VIM-like enzymes (n = 4). The RMTases ArmA, RmtB, RmtC, RmtF, RmtG, and RmtB + RmtF were identified among 51.4%, 13.6%, 4.9%, 24.3%, 1%, and 1%, respectively. Plasmid co-localization of the carbapenemase and the RMTase encoding genes was found among ca. 20% of the isolates. A high diversity was identified in terms of the nature of associations between RMTase and carbapenemase-encoding genes, of incompatibility groups of the corresponding plasmids, and of strain genetic backgrounds, highlighting heterogeneous importations rather than clonal dissemination.