So, they call it 10 "vacation days" specifically, not just PTO. At some jobs, you have a PTO bank that you use for everything - sick, vacation, personal time etc. At others, you have those days designated differently and given/accrued separately. At this site, they referred to them as vacation days, and went out of their way to say that interns do not have sick or personal time.
And yeah, your final set of questions is what I'm getting at! Seems like a stupid policy to me. Between post-doc interviews, defending my dissertation, doctor appointments, and possibly being sick... I don't think 10 days off total for an entire year is going to cut it.
EDIT: They said that the vacation days *needed* to be scheduled at least a month in advance... so they wouldn't be very useful as sick days.