I'm running Ubuntu on an IBM thinkpad (T43).

I've been using VLC to play .avi files, but it seems that sometimes the audio is present, sometimes not. Usually, if I try to play a file after booting up, the audio (and visual) is fine, but if I try again later it has disappeared. The only way to get it back so far is to reboot.

I tried playing the .avi files with Kaffeine, which informed me:
All Audio Drivers failed to initialize

Trying to play them with Movie Player always results in no sound and an extremely slow frame rate as well.

So, it seems that something goes wrong with the audio drivers at some point -- but how do I diagnose this problem? How can I use Ubuntu to inspect the state of the audio drivers? Or is it a different problem?

Ubuntu 8.04?

If so then it is more likely to be a problem with pulseaudio than with the audio drivers. You are not alone.
You can try to switch to OSS or ALSA in system - preferences - sound.

good luck.