Servers are configured to operate in two or more threshold security planes with each such threshold security plane implementing at least a portion of a corresponding threshold security protocol involving at least a subset of the servers. The servers are implemented on at least one processing device comprising a processor coupled to a memory. Multiple ones of the servers may be implemented on a single processing device, or each of the servers may be implemented on a separate processing device. At least one of the servers may be part of at least two of the threshold security planes. A given request for a protected resource is processed through each of the planes in order for a corresponding user to obtain access to the protected resource. By way of example, the security planes may comprise two or more of an authentication plane, an access control plane and a resource plane.