Are you absolutely sure you are using ALSA directly and not OSS-emulation? Very rarely does software automatically use ALSA. If you are using ALSA, which ALSA device are you using? default? hw:0,0?
Some sound cards don't support hardware mixing (playing multiple streams at the same time through hardware). My USB sound devices are like that. For those rare cases, you need to use a dmix device (software mixing) in ALSA.
Unfortunately, that usually boils down to putting some cryptic statements into /etc/asound.conf. If possible, it should be avoided.