Reliability analysis and probability methods are of extreme importance for understanding the behavior of critical systems. In this scope, dynamic fault trees (DFTs) are consolidated graphical models, previously applied at known entities, such as NASA and TESLA. However, the (classical) DFTs analysis has a known issue; the fact that it assumes that the distribution of basic events (BEs) follows the exponential/Weibull distributions, which is often a rough approximation of real data distributions. Moreover, building a DFT model for a real system requires specialized knowledge, to infer the root causes of failures. In this work, a new formalism for the analysis of so-called generic fault trees (GFTs) is proposed which extends DFT to data-driven scenarios, based on the notion of h-approximation of the associated distributions, where leaves may contain an arbitrary compact support distribution or an h-truncation of a distribution. The approach is validated against known solutions in the literature, showing great accuracy. An optimization process is employed to generate the best structure for training a GFT that fits a given data and tested in real use case scenario of a stamping press of Bosch ThermoTechnology. The obtained GFT demonstrates a good fit, considering 3 different metrics.