In most cases the site was performing as expected using the allegedly broken theme, however the custom template selection was no longer available on the Pages admin page. (This is where I first spotted there was a problem and then discovered the Appearances->Themes admin page.)
After poking about for a bit, I traced the problem to a rather innocuous looking line the designer had added to the theme's style.css file.
/* UNIQUE CATEGORY TEMPLATE: VIDEO*/
It's just the "TEMPLATE:" portion that caused the style.css parser to fail. This line works just fine:
/* UNIQUE CATEGORY TEMPLATE : VIDEO*/
I doubt this error comes up often, but it took a few minutes to track down the problem so I thought I'd post something here to hopefully save somebody time in the future.