I'm aware of the damage that can be caused to a person's ears if exposed to frequent amounts of loud sound, but I was wondering if headphones can be damaged by a similar thing. If audio levels are set too high, is it possible to damage the speakers within a pair of headphones?
Yes, it is certainly possible to damage the transducers within a pair of headphones by setting the audio too high.
Were you to attach your headphones to a supply of 240v AC at 50Hz you would be likely to burn out the transducers in a very short space of time.
Under no circumstances at all do I recommend doing this, but that would certainly fall under the banner of 'damaged'. Was that what you had in mind?
Yes, and especially with inear phones, make sure you put them well into your ears. With overhead headphones it's less a problem.
When you don't put inear phones in your ears, most of the sound (especially bass sounds which use more power) will sound much lower, so turning up the volume is the intuitive thing to do, resulting in too much power getting into the inear phones, blowing up the speaker.