Allowing read operations to return stale data with low probability has been proposed as a means to increase availability in quorums systems. Existing solutions that allow stale reads cannot tolerate an adversarial scheduler that can maliciously delay messages between servers and clients in the system and for such a scheduler existing solutions cannot enforce a bound on the staleness of data read. This paper considers the possibility of increasing system availability while at the same time tolerating a malicious scheduler and guaranteeing an upper bound on the staleness of data. We characterize the conditions under which this increase is possible and show that it depends on the ratio of the write frequency to the servers' failure frequency. For environments with a relatively large failure frequency compared to write frequency, we propose K-quorums that can provide higher availability than the strict quorum systems and also guarantee bounded staleness. We also propose a definition of k-atomicity and present a protocol to implement a k-atomic register using k-quorums.