Timing belts and chains are susceptible to the same forces and suffer the same stresses. Either one can break when the tension is greatest – at startup or shutdown of the engine, or at acceleration or deceleration. For rubber timing belts and non-interference type engines, you can usually just install a timing belt kit to maintain the system.
If you have an interference engine, there is going to be some damage no matter at what point in the engine’s operation the belt breaks. How bad that damage is will depend on the engine speed at the time the belt snaps. If it breaks at startup or shutdown, there will probably be some bent or broken valves and valve guides. If the belt breaks while you’re speeding down the highway at high rpm, then there will be more catastrophic damage – valves will shatter and bounce around inside the cylinders, bending the connecting rods and destroying the pistons. As the pistons and rods disintegrate, the damage will expand to the oil pan, the engine block, and other parts of the engine, basically cutting holes throughout the other systems. This sounds bad, and it is, because damage of this magnitude can’t generally be repaired and needs replacement.