We introduce a family of sequential test procedures in the context of a futility study design, or as we prefer to call it, a formal test of promise, suitable for use with time-to-event data. The procedures are motivated by an actual trial that was undertaken to test the promise of very early antiretroviral therapy to achieve viral remission in infants with perinatally-acquired HIV. Important gains in efficiency are illustrated in terms of early stopping and statistical power compared with other methods such as Simon’s two-stage design with binary outcomes. We show how to calculate the operating characteristics of the proposed sequential tests of promise and provide optimal or near-optimal boundaries for small or medium size samples which provide the typical context for the tests under consideration. The design features discussed in this article are also of immediate pertinence to trials designed to test disease cures which may require treatment interruption and small numbers of participants.