Diversity is expected to increase the resilience of ecosystems. Nevertheless, highly diverse ecosystems have collapsed, as did Lake Victoria's ecosystem of cichlids or Caribbean coral reefs. We try to gain insight to this paradox, by analyzing a simple model of a diverse community where each competing species inflicts a small mortality pressure on an introduced predator. High diversity strengthens this feedback and prevents invasion of the introduced predator. After a gradual loss of native species, the introduced predator can escape control and the system collapses into a contrasting, invaded, low-diversity state. Importantly, we find that a diverse system that has high complementarity gains in resilience, whereas a diverse system with high functional redundancy gains in resistance. Loss of resilience can display early-warning signals of a collapse, but loss of resistance not. Our results emphasize the need for multiple approaches to studying the functioning of ecosystems, as managing an ecosystem requires understanding not only the threats it is vulnerable to but also pressures it appears resistant to.