Deep-subwavelength all-dielectric composite materials are believed to tightly obey the Maxwell Garnett effective medium theory. Here, we demonstrate that the Maxwell Garnett theory could break down due to evanescent fields in deep-subwavelength dielectric structures. By using two- and three-dimensional dielectric composite materials with inhomogeneities at a scale of $\lambda /100$, we show that local evanescent fields generally occur near the dielectric inhomogeneities. When tiny absorptive constituents are placed there, the absorption and transmission of the whole composite will show strong dependence on the positions of the absorptive constituents. The Maxwell Garnett theory fails to predict such position-dependent characteristics because it averages out the evanescent fields. By taking the distribution of the evanescent fields into consideration, we have made a correction to the Maxwell Garnett theory so that the position-dependent characteristics become predictable. We reveal not only the breakdown of the Maxwell Garnett theory, but also a unique phenomenon of “invisible” loss induced by the prohibition of electric fields at deep-subwavelength scales. We believe our work promises a route to control the macroscopic properties of composite materials without changing their composition, which is beyond the traditional Maxwell Garnett theory.