One of the main challenges for future wireless technologies is supporting a massive amount of machine-type devices. In the analysis and design of such systems (e.g. internet of things (IoT)), a basic premise is often that devices are acting indedpendently. In a number of practical IoT scenarios, such as distributed sensor networks, information is inherently correlated due to a commonly observed physical phenomenon. In this report we consider a model that includes correlation both in device activation and in the message content. To this end, we introduce a common physical phenomenon that can trigger an alarm causing a subset of devices to transmit the same message at the same time. We develop a new information-theoretic error probability model that includes false positive errors, resulting from decoding particular non-transmitted codewords. The results show that the correlation allows for high reliability at the expense of network spectral efficiency. Additionally, non-orthogonal access with superposition encoding can be preferable to orthogonal access when multi-access interference is low to moderate.