"As for the CD or MP3 file, they're gone as soon as one piece of data is corrupted."
Not necessarily. An MPx file (there are three different audio layers of the MPEG-1 standard that are commonly in use, 2 and 3 being the most prevalent) is really nothing more than a series of "frames" in which audio data are stored in. Think of it in terms of movie film: if a single frame is cut out of a movie and the two ends of the filmstrip are spliced together, the single frame will be gone but, barring mechanical issues with the projector (and the abilities of the dude doing the splicing) you will still see the rest of the movie. So if part of an MPx audio file is missing or corrupted, in most cases, the file still continues playing. You might either hear a "chirp" or a "pop" or whatever wherein the missing data were.
In fact, this is how (why) DVB audio (usually MPEG1 layer 2 based, if it isn't using AC3 which I don't know much about) and MP3 streaming works and why you can generally begin listening at any point in the broadcast. Also why if you stop coding an M3 file about halfway through, you can still (in many cases) play back what you have, but it stops where you stopped coding it.
As for CDs: by my experience it depends on the quality of the drive and software/firmware etc. you are using to play back on. I have noticed some drives and programmes handle disc errors like scratches better than others.
(And here is another thing to remember: contrary to popular belief, if you store CDs loose, it is actually safer to set them down with the playing surface facing down, like you'd insert it into the drive. The data layer on a CD is just a few microns underneath the top of the labelled surface. So if there's something abarasive on the surface you set the disc on and silvering gets beaten up or scratched, the disc can get ruined. You can often buff out scratches in the bottom surface and have the disc still be playable--not so in the top surface!)