Numerous systems in the climate sciences and elsewhere are excitable, exhibiting coexistence of and transitions between a basic and an excited state. We examine the role of tipping between two such states in an excitable low-order ocean model. Ensemble simulations are used to obtain the model's pullback attractor (PBA) and its properties, as a function of a forcing parameter [Formula: see text] and of the steepness [Formula: see text] of a climatological drift in the forcing. The tipping time [Formula: see text] is defined as the time at which the transition to relaxation oscillations (ROs) arises: at constant forcing this occurs at [Formula: see text]. As the steepness [Formula: see text] decreases, [Formula: see text] is delayed and the corresponding forcing amplitude decreases, while remaining always above [Formula: see text]. With periodic perturbations, that amplitude depends solely on [Formula: see text] over a significant range of parameters: this provides an example of rate-induced tipping in an excitable system. Nonlinear resonance occurs for periods comparable to the RO time scale. Coexisting PBAs and total independence from initial states are found for subsets of parameter space. In the broader context of climate dynamics, the parameter drift herein stands for the role of anthropogenic forcing.