Context. The possibility of identifying co-natal stars that have dispersed into the Galactic disc based on chemistry alone is called strong chemical tagging. It has been debated for a long time whether this is indeed feasible; it holds the promise of reconstructing the detailed star formation history of a large fraction of stars in the Galactic disc.Aims. We investigate the feasibility of strong chemical tagging using known member stars of open clusters.Methods. We analysed the largest sample of cluster members that have been homogeneously characterised with high-resolution differential abundances for 16 different elements. We also investigated the possibility of finding the known clusters in the APOGEE DR16 red clump sample with 18 chemical species. For both purposes, we used a clustering algorithm and an unsupervised dimensionality reduction technique to blindly search for groups of stars in chemical space.Results. Even if the internal coherence of the stellar abundances in the same cluster is high, typically 0.03 dex, the overlap in the chemical signatures of the clusters is large. In the sample with the highest precision and no field stars, we only recover 9 out of the 31 analysed clusters at a 40% threshold of homogeneity and precision. This ratio slightly increases when we only use clusters with 7 or more members. In the APOGEE sample, field stars are present along with four populated clusters. In this case, only one of the open clusters was moderately recovered.Conclusions. In our best-case scenario, more than 70% of the groups of stars are in fact statistical groups that contain stars belonging to different real clusters. This indicates that the chances of recovering the majority of birth clusters dissolved in the field are slim, even with the most advanced clustering techniques. We show that different stellar birth sites can have overlapping chemical signatures, even when high-resolution abundances of many different nucleosynthesis channels are used. This is substantial evidence against the possibility of strong chemical tagging. However, we can hope to recover some particular birth clusters that stand out at the edges of the chemical distribution.Read less <