Tipping points, at which complex systems can shift abruptly from one state to another, are notoriously difficult to predict. Theory proposes that early warning signals may be based on the phenomenon that recovery rates from small perturbations should tend to zero when approaching a tipping point; however, evidence that this happens in living systems is lacking. Here we test such 'critical slowing down' using a microcosm in which photo-inhibition drives a cyanobacterial population to a classical tipping point when a critical light level is exceeded. We show that over a large range of conditions, recovery from small perturbations becomes slower as the system comes closer to the critical point. In addition, autocorrelation in the subtle fluctuations of the system's state rose towards the tipping point, supporting the idea that this metric can be used as an indirect indicator of slowing down. Although stochasticity prohibits prediction of the timing of critical transitions, our results suggest that indicators of slowing down may be used to rank complex systems on a broad scale from resilient to fragile.