A PSU that isn't up to the task is not going to give you very steady voltage, with dips and spikes whenever something power intensive happens (loading a game, a drive spinning up, fans kicking in, those 3 other cores revving up to full usage, etc.)

These dips and spikes put extra strain on your components and shorten their lifespan. I doubt a PSU is going to outright fry a component due to being too small, but it might make the difference between a GPU lasting your years or a week after the warranty runs out.

Also don't go with some no-name brand. Some of them are cheap, use really substandard components and CAN fry a system if they spike too high.