Recently, we saw the emergence of consensus-based database systems that promise resilience against failures, strong data provenance, and federated data management. Typically, these fully-replicated systems are operated on top of a primary-backup consensus protocol, which limits the throughput of these systems to the capabilities of a single replica (the primary). To push throughput beyond this single-replica limit, we propose concurrent consensus. In concurrent consensus, replicas independently propose transactions, thereby reducing the influence of any single replica on performance. To put this idea in practice, we propose our RCC paradigm that can turn any primary-backup consensus protocol into a concurrent consensus protocol by running many consensus instances concurrently. RCC is designed with performance in mind and requires minimal coordination between instances. Furthermore, RCC also promises increased resilience against failures. We put the design of RCC to the test by implementing it in ResilientDB, our high-performance resilient blockchain fabric, and comparing it with state-of-the-art primary-backup consensus protocols. Our experiments show that RCC achieves up to 2.75x higher throughput than other consensus protocols and can be scaled to 91 replicas.