We address the problem of optimal downlink resource allocation in an OFDMA system, in a scenario where very limited channel quality information (CQI) is available at the base station. This paper is particularly applicable in the context of the LTE downlink since the feedback mechanism that we consider closely resembles one of the CQI reporting modes in LTE. Specifically, the users only report the indices of their best $M$ subbands and an effective CQI corresponding to these best $M$ bands. Our policy simultaneously performs optimal subband assignment and rate allocation, by taking into account channel quality and the queue backlogs of each user. The technical novelty of our work lies in exploiting a limit theorem on the best SNRs reported by the users, and combining it within a Lyapunov stability framework. We show that our policy is throughput maximizing among all policies, which are constrained to the CQI mechanism considered. Numerical results indicate that, in terms of throughput and average delay, our policy compares favorably to existing resource allocation policies such as proportional fair. © 2016 IEEE.