So, as it turns out, as soon as a character loses relevance to quests, they can die. I finished the Companion quests, and after losing Lydia, the housecarl, to bandits in Swindler's Den, I took Vilkas along for a stroll. Man, he took on a dragon while I fought its brother. I can hear him now... "Today is the day you die, dragon!"
He survived that encounter. Then I made the mistake of investigating the legend of Red Eagle. Turned out Red Eagle didn't want to be woken up, and while I cleared the room of skeletons, he took his hangover out on poor Vilkas.
I don't actually know how it works. Can they be knocked over a certain amount of times before it's fatal, or do certain enemies just attack them while they're down to kill them?
I actually don't want any more Companions to accompany me, in case they all die and can't give me quests anymore.