The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. The authors show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis -- which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines.