We gave NFP a shot; I don't get too many opportunities to share my real experience with it, which is a shame. A little over a year ago my wife was sick of her birth control (subdermal implant), and being from a very Catholic family she wanted to at least learn about and try NFP. I reluctantly agreed (we were stable enough that starting our family a bit earlier than planned wouldn't have been a disaster).
We signed up for a class through SymptoPro through our local church, an organization that provides workshops and materials about it. In summary, I'll the say method is definitely more scientific than old "rhythm" methods, but still leaves a lot to be desired. According to their materials (and duly backed up by studies, as far as I can tell), time between menstruation and ovulation can be quite variable (hence failure of rhythm methods), but time after to the next menstruation is be fairly predictable for a given woman, and further the time of ovulation should be roughly predictable from physical symptoms, and thus also the likely times of fertility, taking into account other factors such as sperm lifetime etc. (the materials are adamant they don't try to predict ovulation per se).
The rules are very complex, but the basic "intro setup" is this: Each day the woman should take her basal body temperature. This must be done in the morning right after waking (ideally still in bed) to avoid noise from other activity. [Work swing shift? Have an over- or under- active immune system? Good luck with that.] Each day she should evaluate her vaginal discharge (character and amount), and for extra accuracy also her cervical opening (yup, you read that right). Each cycle, the last day of these fertile "signs" is taken to be "peak day" (I presume correlated with ovulation, but they don't come out and say it directly). After this peak day is met, she looks for three temperature readings all higher than the previous six; if the third is 0.4 degrees F higher than the highest of the 6, then you're good to sex it up that evening (assuming avoiding pregnancy). There are special cases involved if the third day doesn't reach the 0.4 above level, etc. There are also more advanced, less conservative rules available once 6 to 12 months of individual data have been gathered. It's really the kind of thing that should be handled by software, because it's so difficult to remember and apply. (I could also see it being fertile ground for software bugs. Pun intended.)
Is it scientific, and is it effective? I'll say "kinda" to the first, and "mostly" to the second. There have indeed been studies on pregnancy rates, and results seem comparable to condoms (though condoms have come a long way; high-90s% is about what is claimed by symptopro for perfect use; results vary across studies, YMMV, etc.) What bothers me is that I have yet to identify any science behind the rulesets themselves. Why three days after six, and where does this 0.4 degrees magic number come from? I'd love to be proven wrong, but I think these are essentially someone's hunch.
As to effectiveness, I think it's reasonably effective primarily because it's so conservative. According to the rules, we got on average maybe 6 days per month we *could* have sex and avoid pregnancy (aside from menstruation days); a couple of months we had 0 available days. We ended up using condoms quite a bit anyway. On the plus side, via all this charting we learned that this isn't quite normal: many women get closer to 10 days/cycle of infertile time (even with the basic ruleset), and my wife may have a mild "luteal phase defect." Now that we're actively trying to start our family, we'll be talking to a doctor about it.
In the end, I'd say it's not a terrible program and it's nice to have options. But, more research is needed, and it's far from the easy, one-size-fits-all solution NFP proponents tout.