Media player, QuickTime, and a few others may tell you that they're saving something to your machine when they actually only save a link to the source. You frequently do need to "be connected" in order to play what you think has been saved on your machine - and it can be tough to tell where the "object" is living.
The error messages you got are telling you that you have to get your song from the server each and every time you play it, and that the server is busy - or down - now. All you put on your desktop was a link to the server.