Is it possible for a human father with green eyes and a human mother with blue
eyes to produce a brown eyed child? And if so, what are the chances?

Funny, I have friends who both had blue eyes produce a brown eyed
child.  I always thought it was impossible and she cheated on him. 
She assured me it was his.  I just some things you learn in school is
a lie.

Here is more on the issue:

http://seps.mgd-colo.peak.org/cvoracle/faq/eyecolor.html

Its not so much that you were lied to, just given an
oversimplification. Eye color, like many traits, is controlled by many
genes (polygenic) that can interact in a variety of ways. In addition
to the actual pigmentation, there are also genes that determine the
distribution of pigment (even, blotches etc.) A couple of sites that
provide a bit more information are given below

http://www.athro.com/evo/inherit.html
http://www.thetech.org/genetics/ask.php?id=2

there was a question ver like this posted before, maybe if you look in
the archives you can get your answer

http://answers.google.com/answers/threadview?id=464871

This may be the question Anthy means.

check this out, It's called the eye calculator.  you can enter in your
own allele frequency, and other variables.

http://museum.thetech.org/ugenetics/eyeCalc/eyecalculator.html

it gives exact percentages, based upon your desired information.  enjoy