We show how testing whether a system is diagnosable can be reduced to the satisfiability problem and how satisfiability algorithms can yield a very efficient approach to testing diagnosability. Diagnosability is the question whether it is always possible to hnow whether a given system has exhibited a failure behavior. This is a basic question that underlies diagnosis, and it is also closely related to more general questions about the possibility to know given facts about system behavior. The work combines the twin plant construct of Jiang et al., which is the basis of diagnosability testing of systems with an enumerative representation, and SAT-based techniques to AI planning which form a very promising approach to finding paths in very large transition graphs.