The stability of the synchronized state of coupled time-varying systems can be predicted with good accuracy for noiseless identical oscillators. Such an assumption, however, is unrealistic for real systems where small heterogeneity and noise are always present. In the case of chaotic oscillators, small heterogeneities and/or noise can generate, due to the existence of unstable sets embedded in the attractor, severe desynchronization bursts, known as bubbling, even in the coupling regions that standard linear stability analysis identifies as stable. Here we analyze the bubbling phenomenon in coupled Rössler and Lorenz chaotic oscillators and predict when bubble-free behavior is expected. We present several different criteria, identifying the one that provides the closest prediction of the minimum coupling strength required for high-quality synchronization during a finite-time simulation.