These capacitors from Kemet have flexible termination.
The datasheet says it caters for applications where the board flexes and normal ceramic caps can fail. However, I fail to see a situation where PCB's flex? If PCB's had the potential to flex, wouldn't you just get a tougher enclosure?
Flexible-termination caps are commonly used in automotive applications, where vibration causes the board to flex back-and forth a small amount repeatedly. Normal components in this application can sprout fatigue stress-fractures that gradually enlarge over time until they fail, which in some automotive applications can have catastrophic effects.
It is very hard to completely remove vibration-caused flex without massively supporting a PCB, and that is very expensive, even compared to fancy caps like the Kemet ones linked.
As tronixstuff mentioned, they are also commonly used in situation where the environment is fairly extreme, as there is no way to completely prevent thermal stresses, since the different aspects of a PCB have different coefficients of expansion as the temperature changes.
Basically, they're for high-reliability/extreme-environment applications.
Data sheet says shear stress and thermal cycling. Good points. Furthermore, a PCB may flex in use if it had a socket for something that was inserted and removed more than occasionally. Or perhaps in high-vibration environments such as a data logger in an off-road vehicle?
Here is an example of an interesting case where flexible PCB is used: Mechanical connectors for flexible PCBs?
Like other have said though, the biggest focus is applications where you have vibrations.
Flexing can occur in the manufacturing process for long PCBs. Cracked caps can not all be found in manufacturing tests and could fail in the field.
