In recent years, the Industrial Internet of Things (IIoT) has led to significant steps forward in many industries, thanks to the exploitation of several technologies, ranging from Big Data processing to Artificial Intelligence (AI). Among the various IIoT scenarios, large-scale data centers can reap significant benefits from adopting Big Data analytics and AIboosted approaches since these technologies can allow effective predictive maintenance. However, most of the off-the-shelf currently available solutions are not ideally suited to the HPC context, e.g., they do not sufficiently take into account the very heterogeneous data sources and the privacy issues which hinder the adoption of the cloud solution, or they do not fully exploit the computing capabilities available in loco in a supercomputing facility. In this paper, we tackle this issue, and we propose an IIoT holistic and vertical framework for predictive maintenance in supercomputers. The framework is based on a big lightweight data monitoring infrastructure, specialized databases suited for heterogeneous data, and a set of high-level AI-based functionalities tailored to HPC actors' specific needs. We present the deployment and assess the usage of this framework in several in-production HPC systems.