mp3s are supposed to play even if chopped (i mean, incompletely downloaded).

Use Go!zilla to leech what you need from the web. You gotta get the earlier versions though, like 2.9, so that you wouldn't have to bother with the Gator program ( spy software!) it comes with. Or you could go DAP if you want.
If all else fails, get camstudio 1.8 (another free and opensource program! Your solution to save streamed audio and/or video) and capture the audio while listening to the streamed audio.