We show that a continuous linear operator T on a Fréchet space satisfies the
so-called Hypercyclicity Criterion if and only if it is hereditarily hypercyclic, and if
and only if the direct sum $T \oplus T$ is hypercyclic. In particular, hypercyclic operators with either a dense generalized kernel or a dense set of periodic points (i.e., chaotic in the sense of R. L. Devaney (1989, ``An Introduction to Chaotic Dynamical Systems,'' Addison-Wesley, Reading, MA)) must satisfy the Criterion. Finally, we provide a characterization of those weighted shifts T that are hereditarily hypercyclic with respect to a given sequence $(n_k)$ of positive integers, as well as conditions under which T and $\{ T^{n_k} \}_{k \geq 1}$ share the same set of hypercyclic vectors.