Will the Next Generation Internet provide an appropriate infrastructure for critical applications, such as are emerging in such settings as health care, electric power grid control, air traffic control, banking, and military command and control? This paper suggests that current trends are unlikely to yield the required platform. Drawing lessons from successful critical networking projects of the past, we propose an alternative based on Virtual Overlay Networks (VONs). Support for VONs would not require radical departure from existing router capabilities, because a limited mechanism of the type we propose is already available. Given a simple Overlay Network (ON) capability, we show that tools for building more sophisticated VONs are already widely available.