One of the central questions in distributed computability is characterizing the tasks that are solvable in a given system model. In the anonymous case, where processes have no identifiers and communicate through multi-writer/multi-reader registers, there is a recent topologi-cal characterization (Yanagisawa 2017) of the colorless tasks that are solvable when any number of asynchronous processes may crash. In this paper, we consider the case where at most t asynchronous processes may crash, where 1 ≤ t < n. We prove that a colorless task is t-resilient solvable anonymously if and only if it is t-resilient solvable non-anonymously. We obtain our results through various reductions and simulations that explore how to extend techniques for non-anonymous computation to anonymous one.