Never mind, I fixed it. It turned out to be a permissions issue, which makes no sense at all. If anyone at Adobe is reading this, please tell your dev team that if a user has permission to read a file, they should have permission to play that file. Thanks.
If anyone else has this problem, here's what I did to fix it:
Type the following commands:
chown -Rv yourusername *
chmod -Rv a=rwxrwxrwx *
Enable root user (http://support.apple.com/kb/ht1528)
Log in as root user
Open the project file
Save the project file
Log back in as yourself
It should be fixed.
I would have used the Disk Utility "Repair Permissions" feature, but for some reason it was grayed out for this drive.
Saving the project actually has nothing to do with it - the fact that you logged in as root is what fixed your problem. Check around for other posts about playback problems on mac (or do a search for the keyword 'aggregate') - it has to do with a screwed up state of your audio driver. The less convoluted workaround involves deleting a plist file from your preferences folder.
Fascinating. I did some searching like you said and found this article with instructions: http://kb2.adobe.com/cps/842/cpsid_84287.html
I have no idea why logging in as a new user with elevated permissions would alter the driver configuration, or why Premiere would be able to play sound for a few frames and have the driver kick back some sort of instruction that forces it to stop, but apparently this fixes it.
Thanks for your help Will. As a system administrator, I'd love to learn more about what's wrong with the driver configuration in this case. I don't know if you have that information, but if you do, I'd love to have it.