An isolated quantum many-body system in an initial pure state will come to thermal equilibrium if it satisfies the eigenstate thermalization hypothesis (ETH). We consider alternatives to ETH that have been proposed. We first show that von Neumann's quantum ergodic theorem relies on an assumption that is essentially equivalent to ETH. We also investigate whether, following a sudden quench, special classes of pure states can lead to thermal behavior in systems that do not obey ETH, namely, integrable systems. We find examples of this, but only for initial states that obeyed ETH before the quench.