We present a study of the variations in the absorbing column density of 25 X-ray-defined Seyfert 2 galaxies, as inferred from hard X-ray observations, on timescales from months to several years. We show that a significant variation of NH (from 20% to 80%) is observed in almost all (22 of 25) of the sources with multiple X-ray observations, although X-ray absorption never vanishes. For a subsample of 11 sources observed at least five times, the typical variation time, as defined by a structure function, is less than 1 yr for both heavily absorbed (NH ~ 1023 cm-2) and moderately absorbed (NH ~ 1022 cm-2) sources. These variations rule out the simplest version of the unified models, based on a homogeneous obscuring torus, and suggest the presence of clumpy circumnuclear material on a scale well below a parsec. We propose a modification of the torus model in which an overabundance of slightly dusty broad emission-line region (BELR) clouds obscures the BELR. The BELR needs, like the torus, to have an axisymmetric structure. This model is closely related to that of Elvis for type 1 active galactic nuclei (AGNs). For lightly obscured AGNs (NH ~ 1022 cm-2), the structure function shows an increase at a timescale of ~5 yr, indicating a second absorber, most probably on a 5-10 pc scale associated with the host galaxy.