To achieve high reliability in on-chip networks, it is necessary to test the network as frequently as possible to detect physical failures before they lead to system-level failures. A main obstacle is that the circuit under test has to be isolated, resulting in network cuts and packet blockage which limit the testing frequency. To address this issue, we propose a comprehensive network-level approach which could test multiple routers simultaneously at high speed without blocking or dropping packets. We first introduce a reconfigurable router architecture allowing the cores to keep their connections with the network while the routers are under test. A deadlock-free and highly adaptive routing algorithm is proposed to support reconfigurations for testing. In addition, a testing sequence is defined to allow testing multiple routers to avoid dropping of packets. A procedure is proposed to control the behavior of the affected packets during the transition of a router from the normal to the testing mode and vice versa. This approach neither interrupts the execution of applications nor has a significant impact on the execution time. Experiments with the PARSEC benchmarks on an 8x8 NoC-based chip multiprocessors show only 3 percent execution time increase with four routers simultaneously under test.