In the analysis of cryptographic protocols, a "treacherous" set of terms is one from which an intruder can get access to what was intended to be secret, by adding on to the top of a sequence of elements of this set, a "cap" ormed of symbols legally part of his/her knowledge. In this paper, we give sufficient conditions on the rewrite system modeling the intruder's abilities, such as using encryption and decryption functions, to ensure that it is decidable if such caps exist. The following classes of intruder systems are studied: linear, dwindling, Delta-strong, and optimally reducing; and depending on the class considered, the cap problem (``find a cap for a given set of terms'') is shown respectively to be in P, NP-complete, decidable, and undecidable.