Another big source for error too comes from rounding. We only touched upon this briefly but it's an interesting concept to think about - but thinking out loud here, I guess you can consider something like rounding human error. At first glance, something like rounding may not seem like a huge deal but when you're dealing with the computation of tens of thousands (maybe even in the hundredths of thousands or millions?) of equations, errors due to rounding are going to add up quickly and this could very well be a large source for error when you start getting out to say 3 days.
You could also argue that modeling is rather accurate out to 15-16 days because when you're getting that far out in time what are your two options really? Modelling and climatology. I never knew this until this was discussed briefly as well in my class, but the reason the GFS is ran to 384 hours is because that is about the cut-off of when modeling has an advantage over climatology. After that time climatology tends to outweigh modelling. Interesting stuff