Distributed systems such as grids, peer-to-peer systems, and even Internet DNS servers have grown significantly in size and complexity in the last decade. This rapid growth has allowed distributed systems to serve a large and increasing number of users, but has also made resource and system failures inevitable. Moreover, perhaps as a result of system complexity, in distributed systems a single failure can trigger within a short time span several more failures, forming a group of time-correlated failures. To eliminate or alleviate the significant effects of failures on performance and functionality, the techniques for dealing with failures require good failure models. However, not many such models are available, and the available models are valid for few or even a single distributed system. In contrast, in this work we propose a model that considers groups of time-correlated failures and is valid for many types of distributed systems. Our model includes three components, the group size, the group inter-arrival time, and the resource downtime caused by the group. To validate this model, we use failure traces corresponding to fifteen distributed systems. We find that space-correlated failures are dominant in terms of resource downtime in seven of the fifteen studied systems. For each of these seven systems, we provide a set of model parameters that can be used in research studies or for tuning distributed systems. Last, as a result of our work six of the studied traces have been made available through the Failure Trace Archive (http://fta.inria.fr).