The |$n$|-dimensional burnt pancake network, denoted by |$BP_n$|⁠, originates from the Burnt pancake problem which relates to the construction of networks of parallel processors. In the wake of rapid development of multiprocessor systems, processor fault diagnosis plays an even more important role in measuring the reliability of a multiprocessor system, and the diagnosabilities of many well-known multiprocessor systems have been investigated. The conditional diagnosability has been widely accepted as a new measure of diagnosability by assuming that any faulty set cannot contain all the neighbors of any node in a multiprocessor system. In this paper, we explore combinatorial properties of burnt pancake networks, and investigate the structural vulnerability as well as extra connectivities. Furthermore, we show that the classic diagnosability and the conditional diagnosability of |$BP_n$| (⁠|$n\geq 4$|⁠) under the Preparata, Metze and Chien model are |$n$| and |$8n-13$|⁠, respectively.