Since the advent of ultra-reliable and low-latency communications (URLLC), the requirements of low-latency applications tend to be completely characterized by a single pre-defined latency-reliability target. That is, operation is optimal whenever the pre-defined latency threshold is met but the system is assumed to be in error when the latency threshold is violated. This vision is severely limited and does not capture the real requirements of most applications, where multiple latency thresholds can be defined, together with incentives or rewards associated with meeting each of them. Such formulation is a generalization of the single-threshold case popularized by URLLC and, in the asymptotic case, approximates to defining a cost for each point in the support of the latency distribution. In this paper, we explore the implications of defining multiple latency targets on the design of access protocols and on the optimization of repetition-based access strategies in orthogonal and non-orthogonal multiple access scenarios with users that present heterogeneous traffic characteristics and requirements. We observe that the access strategies of the users can be effectively adapted to the requirements of the application by carefully defining the latency targets and the associated rewards.