That the cerebral cortex processes information at prodigious speeds cannot be doubted. Yet the passive time constant, tau(m), of neurons, often thought of as a measure of the neuron's "response time' to synaptic input, is relatively long. In the 1950s, tau(m) was estimated to be only a few milliseconds for mammalian central neurons; with improvement in recording techniques, its estimated value grew over the years and it now stands near 20-100 msec. However, as we will argue here, the functional meaning of tau(m) is ambiguous. On the basis of a newly introduced definition of local delay, we show that the time window for synaptic integration in passive dendritic trees can be much smaller than the time constant. We argue that the voltage response to very brief synaptic inputs is essentially independent of tau(m). We discuss how tau(m) can change dynamically with the global activity of the network, as well as the difficulties of defining a time constant in structures with voltage-dependent elements. We conclude that the classically defined tau(m) only provides a very rough estimate, typically an overestimate, of the response time of neurons and that alternative measures are required to capture the dependency of the time course of the membrane potential on ligand-gated and/or voltage-dependent membrane conductances.