Could be either one! But it sounds most likely to be low specificity -- that tends to be what gives lots of false positives. Or rather, what fails to give true negatives -- there is some tedious distinction there I don't remember...

I just found myself explaining sensitivity and specificity to Graham the other day and he really disapproved of the word specificity in this context. I say someone who studied math (like he did) has no grounds to complain about ordinary English being repurposed as jargon. :)