Counterexamples are presented to weighted forms of the Weiss conjecture in discrete and continuous time. In particular, for certain ranges of a, operators are constructed that satisfy a given resolvent estimate, but fail to be delta-admissible. For alpha is an element of (-1, 0) the operators constructed are normal, while for alpha is an element of (0, 1) the operator is the unilateral shift on the Hardy space H(2)(D).