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In networked systems, the interplay between the dynamics of individual subsystems andtheir network interactions has been found to generate multistability in various contexts.Despite its ubiquity, the specific mechanisms and ingredients that give rise to multistabilityfrom such interplay are still poorly understood. In a network of coupled excitable units, weshow that this interplay generating multistability occurs through competition between theunits' transient dynamics and their coupling. Specifically, the diffusive coupling between theunits manages to reinject them in the excitability region of their individual state space andeffectively trap them there. We show that this trapping mechanism leads to the coexistenceof multiple types of oscillations: periodic, quasiperiodic, and even chaotic - even though theunits do not exhibit oscillations individually. Interestingly, we show that the attractors emergethrough different types of bifurcations - in particular, the periodic attractors emerge througheither saddle-node bifurcations of limit cycles or homoclinic bifurcations - but in all cases thereinjection mechanism is present.