We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems, in particular, for chaotic systems, we argue that the expectation value of a local operator averaged over a time interval of length T is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium of all states with a given value of energy fluctuations. This result applies to any initial state with a well-defined effective temperature. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at a macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with the transport and the properties of microscopic matrix elements. The form of the deviation function indicates that among the "slowest" states which saturate the bound there are also those with arbitrarily long equilibration times.