In the few large company environments I've worked at, banking up unused vacation days was actively discouraged; it's an accounting problem (accrued days have to be paid on separation). That, plus the fact that virtually every established company has policies restricting year-to-year carryover of vacation, suggests that the "culture of presenteeism" idea, while compelling, is not actually suppressing actual vacations.
I'm not so sure the answer is automatically no - I think a lot of workers will save a few vacation days as a buffer in case they need them for something unexpected.
Also, the figure is probably skewed by companies that won't let the leave count go negative - in those cases it probably won't balance out to zero unless the employee makes a specific effort to.
Yeah, the idea of a fixed number of sick days is baffling to me as a European. It's like you're penalized for not being sick, by getting less time off...
From the article: "In 2008 it was estimated that three vacation days were left unused for most American workers."
Since the average American doesn't use all the vacation he has, the answer is probably no.