I think you are right to be anxious about the possibility of a false-positive (the test says there is a problem when there isn't). The NST has a high false-positive rate in general and, according to this systematic review (a structured study of studies on a particular topic), the NST has not been shown to be beneficial in high- or intermediate-risk pregnancies. Moreover, the NST looks for fetal heart rate response to such things as Braxton-Hicks contractions and fetal movement, and somewhere rattling around in my brain, I recall that until the fetus reaches maturity, the fetal heart rate doesn't respond to these sorts of stimuli. Don't quote me on that, though, because I might be remembering wrong. I suggest you ask your dr to show you research confirming the benefits of weekly NSTs because they certainly aren't harmless. A test result saying the baby may have a problem can't be ignored, which is likely to lead to an unnecessary induction, cesarean, and possibly a preterm delivery. If your dr can, then you will have to weigh the benefits against the risks of a false positive, but if your dr can't, then you may wish to decline.
-- Henci