I do, all the time. If you don't pay attention to such things you are in for a surprise. TVS diodes are an excellent example. This here diode is capable of absorbing a 200W pulse! But the diode is only 1x2 mm in size! Must be a miracle, right?
No, sadly it's not a miracle. The diode is capable of clamping that much current only for a few microseconds. Then those 200W of power heat the junction up, and the part goes up in smoke. In this case, 200W * 20 µs = 4 mJ. From that we can calculate the final temperature rise as dT = Q/mC. For example, the mass of the die in this part is 1 mg, and C=0.7J g-1°C-1, so dT = 4*10-3/(1*10-3 * 0.7) = 5 °C. That's a lot for a pulse that you can't see with a multimeter! If you look into the datasheet you will see more complexity because not all pulses are the same; and there is also the bonding wire that limits the current... the manufacturer has to calculate its temperature rise before it melts.
Power means not that much here unless you fully understand the energy that the component has to absorb - and what happens to it (conductive heating of the part, of the board, of the heatsink, radiation, convection.) Some larger parts have heatsinks, and you have to calculate how they help you in dealing with longer pulses. That's when thermal resistances come into play, and specific heat of all materials that are involved... sometimes you end up using CFD tools to validate your design.