In regression, is functional confidence bound influenced by the std err of each yi?

In regression, is functional confidence bound influenced by the std err of
each yi?

I am trying to fit {(xi,yi)} to a curve y = f(x) + ep(x).
Each yi has std err SEi. I use weighted regression, wi = 1/SEi^2/Ni, where
Ni is the number of measurements used to get yi.
If I amplified each SEi by a factor of 10, the functional confidence band
of the fitted curve remains the same, because the relative weights are
still the same. However, according to intuition, the functional confidence
band should be greater now since the uncertainty SEi of each data point is
greater.
Could someone explain this?
Thanks!