In this paper we analyze the promising, but yet mostly unexplored, ability of opportunistic spectrum access (OSA) based on cognitive radios (CR) to provide a robust infrastructure for wireless networks operating in challenging environments with frequent transmission link disruptions. We consider a general network model where the CR users have the capabilities of spectrum sensing and channel switching to determine the status of a channel and use a restoration process when a link failure occurs. We first propose a classification of the reliability metrics in CR networks based on the perspective and severity of the failures. We then derive analytical relations for the mean time to failure (MTTF) and mean time to repair (MTTR) of the CR users. With the proposed OSA channel restoration scheme, we show that the MTTF between hard failures, where a user can not communicate for a long interval, increases exponentially with the number of channels available to the CR users. When a failure occurs, the MTTR also exponentially decreases with the number of channels, thereby providing a highly-robust communication environment. Finally, we provide design guidelines which can be used to evaluate the tradeoffs between the number of users and channels, versus the required reliability.