As I understand it the reason for the impulse test is to check how the device handles short overvoltages over the insulation barrier. Is it possible/likely to fail the impulse withstand if the device can take a dielectric strength test at the same voltage as the impulse voltage. In other words, does the rapid rise time of the impulse withstand cause more problems than just than a static voltage, with ramp up, of the same level?
It can do. It depends on the failure modes of the device, and how it has been built to withstand the high voltage.
Consider a string of insulators. You would hope that the input voltage is distributed uniformly down the string, so that in a string of N insulators, each sees 1/Nth of the applied voltage. Perhaps the insulators are each linked across by a large resistor, so that differences in their leakage current are swamped by the current through the resistors. At DC, these resistors will maintain a uniform voltage gradient down the string.
Now suppose the lower few insulators have an excess stray capacitance to ground. A rapidly rising pulse will charge these lower insulators less than the upper ones. In the time before the resistors have had a chance to equalise the voltages, the higher voltage on the upper insulators could cause them to fail.
While I'm answering, I'll add that this sort of situation is why the Dielectric Strength Test is done over an extended time. If the same string of insulators relied on their equal capacitances to distribute the voltages nicely during the application of the high voltage, but the resistors were absent or too big to maintain equal voltage in the event of unequal leakage currents through the insulators, then eventually the voltage across some insulators could drift up to breakdown levels.
