Context. The [Mg/Fe] abundance ratios are a fundamental fossil signature used to trace the chemical evolution of the disc and to divide it into low-α and high-α populations. Despite the huge observational and theoretical efforts, discrepancies between models and data are still present and several explanations have been put forward to explain the [α/Fe] bimodality. Aims. In this work we take advantage of a new AMBRE:HARPS dataset, which provides new more precise [Mg/Fe] estimations and reliable stellar ages for a subsample of stars, to study the [α/Fe] bimodality and the evolution of the solar neighbourhood. Methods. The data are compared with detailed chemical evolution models for the Milky Way, exploring the most used prescriptions for stellar yields and different formation scenarios for the Galactic disc (i.e. the delayed two-infall and the parallel models), including prescriptions for stellar radial migration. Results. We see that most of the stellar yield prescriptions struggle to reproduce the observed trend of the data and that semi-empirical yields describe best the [Mg/Fe] evolution in the thick and thin discs. In particular, most of the yields still predict a steeper decrease of the [Mg/Fe] ratio at high metallicity than shown by the data. The bulk of the data are well reproduced by the parallel and two-infall scenarios, but both scenarios have problems in explaining the most metal-rich and metal-poor tails of the low-α data. These tails can be explained in light of radial migration from the inner and outer disc regions, respectively. Conclusions. Despite the evidence of stellar migration, it is difficult to estimate the actual contribution of stars from other parts of the disc to the solar vicinity in the data we adopt. However, the comparison between data and models suggests that peculiar histories of star formation, such as that of the two-infall model, are still needed to reproduce the observed distribution of stars.