Can you play multiple songs by playing a song and then closing and reopening the player? If so, I'd guess that maybe there's a library that WMP and Winamp have in common that's causing the bug. You could try downloading another player and using that, since it might not use the same thing. If not, then it isn't the media player, and the only thing I can think of is the sound driver. You could try getting the latest drivers for your soundcard.
Also, does this affect playing multiple videos, or only multiple sound files?
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"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton