Thanks for the suggestions Maxim. I think there is some progress described below:

Maxim Dounin Wrote:
-------------------------------------------------------
> Probably (at least your installation of) IE6 has
> problems with
> downloading (big?) .js files over keepalive
> connections. Gzip has
> mostly nothing to do with it as 1) it's disabled
> as expected, 2)
> one can't rely on content being gzipped all over
> the net anyway
> and 3) having problems with non-gzipped content
> doesn't mean there
> are no problems with gzipped one.

I'm not sure I understand what you mean. It seems like gzip is involved in the problem, either on IE's end or nginx's end, based on this summary:
With nginx conf file having "gzip off;", the problem exists (for .js files. other files work OK)
With 'gzip on; gzip_disable "msie6"', the problem exists (for .js files. other files work OK)
Keeping "gzip on;" and removing 'gzip_disable "msie6"', fixes the problem for .js files, and other files continue to work OK.

> You may want to check the following:
>
> 1. If size or actual javascript content matters
> (i.e. if using
> smaller or simplier file makes any difference).
Yes! I created a test file with a single javascript command (alert('test message');) followed by a long block comment. When the text file's size was less than 2596 bytes (gzip'd file was 719 bytes), it worked just fine. Any larger, even by a single character, and it would show the same delay as before. What does this mean?

> 2. If removing gzip completely makes any
> difference.
See above. the problem exists when nginx's conf file has "gzip off;" too.

> 3. If using another web server with keepalive
> support makes any
> difference.
I havent tried this

> 4. If your IE6 installation is able to cache files
> (and time is
> set correctly). Note that most of the problems
> with "IE can't
> download file" seems to be caused by "can't save
> to disk but need
> to" problem, see here:
>
> http://support.microsoft.com/kb/812935
This souded like a different problem, involving https. In any case, they recommend unchecking a box to fix it (if I read it correctly), and it was already unchecked in the IE options.