Self-blindable credential schemes allow users to anonymously prove ownership of credentials. This is achieved by randomizing the credential before each showing in such a way that it still remains valid. As a result, each time a different version of the same credential is presented. A number of such schemes have been proposed, but unfortunately many of them are broken, in the sense that they are linkable (i.e., failing to protect the privacy of the user), or malleable (i.e., they allow users to create new credentials using one or more valid credentials given to them). In this paper we prove a general theorem that relates linkability and malleability in self-blindable credential schemes, and that can test whether a scheme is linkable or malleable. After that we apply the theorem to a number of self-blindable credential schemes to show that they suffer from one or both of these issues.