Is it possible that electromigration cause a short circuit in a PCB? Theoretically due to chemical potential difference between gold and nickel (1.68 V) it's known that metal ions can penetrate into the isolation and form a bridge. Is that really a practical threat and such a process is really causing short circuit in PCB devices? Is that process occurring only when the device is turned off? Could that make a short circuit between two close pads of a component for example a smd capacitor 0603?
If you have a mechanism for electromigration, than yes I suppose its possible. Possible mechanisms involve battery-like processes and\or an electric current. There are certain materials that develop "tin wiskers". If you are exposing your PCB to chemicals that could aid in the breakdown of metals and cause corrosion\reformation of a metal. If you have a really high current then this is also possible because you turn the metal into a plasma and it deposits elsewhere, like a sputtering machine. Metals don't move around on there own, they are stable, they need energy to move.
Keep in mind that just because its possible doesn't mean its probable. If you are having issues with shorting in a PCB I would first suspect the manufacturer, get an unpopulated PCB and inspect it. If its on the top layer then it should be easy to spot and measure on an unpopulated PCB. If its an internal plane sand off the problem area very carefully to expose the problem area (you can see through the PCB in a lot of cases by sanding off the solder mask, you can even sand through planes.) If it is a problem then contact the manufacturer.
If your having an issue with a component, then it could be the assembly process. When you get your boards back from the assembly, do a visual inspection. Even better, pull the capacitor off the board and measure the impedance and make sure its open. Then put it back on the board and make sure there are no solder bridges.
If your having a cap short out, if it's a polarized cap, you probably have a polarity problem. Tantalum caps should be rated to ~70% of their working voltage. Ceramic caps voltage rating should not be exceeded. Keep in mind there is also esr in a cap, it is possible to have some self heating depending on the conditions.
It is very unlikely that you would see real electromigration in something as "gigantic" (relatively speaking) as a PC board. We typically speak of electromigration at nearly microscopic dimensions (such as inside an integrated circuit).
OTOH, tin-whiskers are becoming a serious problem with the mandating of lead-free solder. ROHS is creating a whole new class of "planned obsolescence". In fact critical products (medical, aerospace, military, etc.) are exempt from the ROHS lead-free rules specifically because of the tin-whisker problem.