We investigate the time evolution of a bundle of fibers subject to a constant external load. Breaking events are initiated by thermally induced stress fluctuations followed by load redistribution which subsequently leads to an avalanche of breakings. We compare analytic results obtained in the mean-field limit to the computer simulations of localized load redistribution to reveal the effect of the range of interaction on the time evolution. Focusing on the waiting times between consecutive bursts we show that the time evolution has two distinct forms: at high load values the breaking process continuously accelerates towards macroscopic failure, however, for low loads and high enough temperatures the acceleration is preceded by a slow-down. Analyzing the structural entropy and the location of consecutive bursts we show that in the presence of stress concentration the early acceleration is the consequence of damage localization. The distribution of waiting times has a power law form with an exponent switching between 1 and 2 as the load and temperature are varied.