This paper considers the diagnosis of large discrete-event systems consisting of many components. The problem is to determine, online, all failures and states that explain a given sequence of observations. Several model-based diagnosis approaches deal with this problem but they usually have either poor time performance or result in space explosion. Recent work has shown that both problems can be tackled when encoding diagnosis approaches symbolically by means of binary decision diagrams. This paper further improves upon these results and presents a decentralised symbolic diagnosis method that computes the diagnosis information for each component off-line and then combines them on-line. Experimental results show that our method provides significant improvements over existing approaches.