Contact tracing, case isolation, quarantine, social distancing, and other non-pharmaceutical interventions (NPIs) have been a cornerstone in managing the COVID-19 pandemic. However, their effects on disease dynamics are not fully understood. Saturation of contact tracing caused by the increase of infected individuals has been recognized as a crucial variable by healthcare systems worldwide. Here, we model this saturation process with a mechanistic and a phenomenological model and show that it induces an Allee effect which could determine an infection threshold between two alternative states—containment and outbreak. This transition was considered elsewhere as a response to the strength of NPIs, but here we show that they may be also determined by the number of infected individuals. As a consequence, timing of NPIs implementation and relaxation after containment is critical to their effectiveness. Containment strategies such as vaccination or mobility restriction may interact with contact tracing-induced Allee effect. Each strategy in isolation tends to show diminishing returns, with a less than proportional effect of the intervention on disease containment. However, when combined, their suppressing potential is enhanced. Relaxation of NPIs after disease containment--e.g. because vaccination--have to be performed in attention to avoid crossing the infection threshold required to a novel outbreak. The recognition of a contact tracing-induced Allee effect, its interaction with other NPIs and vaccination, and the existence of tipping points contributes to the understanding of several features of disease dynamics and its response to containment interventions. This knowledge may be of relevance for explaining the dynamics of diseases in different regions and, more importantly, as input for guiding the use of NPIs, vaccination campaigns, and its combination for the management of epidemic outbreaks.