Three-dimensional (3D) integration is a key technology for systems whose performance and power requirements cannot be achieved by traditional silicon technologies. 3D chips consist of two or more stacked silicon dies connected by short inter-die wires called Thru-Silicon-Vias (TSVs). Despite its potential, the poor reliability and yield, thermal management and testing issues remain major challenges of 3D integration. We address the TSV interconnect test challenge of 3D chips by using Interconnect Built-In Self-Test (IBIST) techniques. The proposed test strategy must sensitize structural faults like opens and shorts, and delay faults due to crosstalk. A possible approach is the well-known Maximum Aggressor Fault (MAF) model. Unfortunately, this model is too conservative and it leads to long test sequences and non-negligible hardware costs. Therefore, we present an alternative solution: the Kth-Aggressor Fault (KAF) model. In our model, aggressors of victim wires are neighboring wires within an optimized distance order K. The aggressor order K is technology-dependent and is determined such that the test times are minimal and the fault coverage is maximal. KAF-based IBIST implementation targeting TSV tests occupies three times less area than similar MAF-/marching-based implementations. We also propose a reconfigurable KAF-based IBIST implementation where tests can be performed using different aggressor orders K. Although the reconfigurable IBIST area is significant, interconnect tests during system lifetime can be performed using lower aggressor orders, reducing test duration and improving TSV availability.