Some item response theory (IRT) techniques work in applications even though the usual structural IRT assumptions, and local independence (LI) in particular, do not hold. When the departure from LI is too great, traditional procedures will break down. Although violations of strictly unidimensional, monotone, locally independent latent structure can sometimes be modeled and exploited, many situations call for a unidimensional approach that is tolerant of minor violations of strict unidimensionality. Departures from strict unidimensionality can be detected, and the influence of these departures on a variety of LI-based ability estimators can be measured. A convenient universe of models near the LI model in which to investigate structural robustness issues is provided by the essential unidimensionality modeling approach of W. F. Stout (1990). Theoretical results underpinning the approach are surveyed. Work in progress to apply these results in practical settings is described; the goal is to develop guidelines for the detection of departures from unidimensionality. Three data tables and 10 figures illustrate the discussion. A 33-item list of references is included. (Author/SLD)