Certain faults, such as intermittent or non-persistent faults, may occur repeatedly. For discrete-event systems prone to repeated faults, Jiang, Kumar, and Garcia in 2003 introduced the notion of-diagnosability requiring the diagnosis of the lth occurrence of a fault within a bounded delay. The present paper studies the identification of the set of all indices for which the system is not-diagnosable. (These are precisely the occurrence indices for which a repeatable-fault cannot be diagnosed.) We present an algorithm of cubic complexity to determine whether a system is diagnosable for every fault-occurrence index. For those systems for which the answer is negative, we show that the set of indices for which the system is non-diagnosable possesses the property that it is either finite or consists of a finite number of eventually periodic sets.