I'm usually the world's biggest cynic regarding hi-res audio. There are way too many charlatans out there getting rich by exploiting the ignorant - and some of these fakers are even selling fake hi-res files!: http://www.head-fi.org/t/648932/bandcamp-promotion-thread/15#post_13283024
However, just for fun, I'm going to do my usual thing of playing Devil's advocate, for the very specific case of 24-bit. I made a comment on an earlier thread of Eke's, which I'll repeat here.
Imagine you have a great recording with a huge dynamic range, good headphones (obviously, or why would you be on headfi?!) and a good amp with a low noise floor. Now imagine raising the volume (dramatically, if you like) just during the quietest moment of the track. This is the audio dynamic-range equivalent of putting a hi-def picture under a microscope. Your ears still only need to cope with a limited dynamic range - you're simply using your amplifier to lower the gain during the Saturn V rocket launch and then raise it during the string quartet intro. Theoretically, you should then be able to 1) not destroy your hearing and 2) hear the difference between a 24-bit and 16-bit dithered noise floor. This is also relevant to the (rather unhelpful) single THD+N measurements given by amp/DAP manufacturers. I would not assume you won't ever be able to hear those apparently tiny differences, unless you only ever listen to 1 kHz sine waves at 80 dB.
I recently found the following article, which describes this more eloquently than I ever could:
http://www.tonmeister.ca/wordpress/2014/09/15/audio-mythinformation-16-vs-24-bit-recordings/