A common technique widely used to achieve fault tolerance in systolic arrays consists in incorporating in the array additional processing elements (PEs) and extra bypass links. Given a sufficient number of PEs and a large enough set of bypass links, it might seem that the array can easily tolerate a large number of faults provided they do not occur in consecutive locations. It is not always the case as shown in this paper. In fact, certain fault patterns exist and may occur which would prevent any kind of restructuring of the aray, thus making the structure unusable. For a given set of bypass links from each PE in the array, it is possible to identify such fault patterns which will prevent any reconfiguration. In this paper, we identify the class of fault patterns that are catastrophic for linear systolic arrays, examine their characteristics, and describe a method for constructing such fault patterns.