We provide experimental evidence suggesting that the NF-κB transcription factor can multiplex information about changes in multiple signals in the sense that the NF-κB target genes response can identify which of these signals have changed. In view of this, we consider how a signalling system can act as an information hub by multiplexing multiple signals. We formally define multiplexing, mathematically characterise which systems can multiplex and how well they can do it. We believe this may resolve the apparent paradox of how a system like NF-κB that regulates cell fate and inflammatory signalling in response to diverse stimuli can appear to have the low information-carrying capacity suggested by recent studies on scalar signals. In carrying out our study, we introduce new methods for the analysis of large, nonlinear stochastic dynamic models, and develop computational algorithms that facilitate the calculation of fundamental constructs of information theory such as Kullback-Leibler divergences and sensitivity matrices, and link these methods to new theory about multiplexing information. We show that many current models such as those of the NF-κB system cannot multiplex effectively and provide models that overcome this limitation using post-transcriptional modifications.