After another one of my regular multi-boot experiments, Vista became unbootable.  I started writing this post while I expected that I would eventually need to ask here for help so that any helpful 9ers would know what I had already tried, but I eventually managed to fix it myself (see the last two paragraphs for the actual solution to my problem).  Since I'd already written most of this, I decided I may as well just post it anyway and maybe someday someone with a similar problem will come across it and it will help them out.

I tried booting from the Vista DVD and going into the repair options.  There was no Vista installation listed on the first screen, so I just clicked next.  I then tried running the startup repair tool.  It told me that it was attempting to repair disk errors, but after a few minutes it said it could not repair this computer automatically.

I then tried running chkdsk /R at the recovery environment command prompt, and it completed succesfully telling me there were no problems (or it had fixed any that were there).

I then tried running the startup repair tool again, with the same results as before.

I then tried going back to the recovery environment command prompt and ran bootrec /fixboot and it told me "The volume does not contain a recognized file system."  bootrec /scanos found Vista without problems, bootrec /rebuildbcd found Vista, and then gave the "The volume does not contain a recognized file system" error.

I then tried navigating around the C: drive on the command prompt and that worked fine.

I then tried using diskpart and found that the partition was not active, so I made it active and tried bootrec /fixboot again.  This time it said it was successful, so I rebooted the computer but still couldn't boot Vista.

I went back to the recovery environment and this time it was able to find the Vista installation and before I could even try anything, it came up with a message saying that it had detected some startup problems and did I want it to automatically repair them and restart.  I told it to go ahead and this time it was able to boot Vista fine.