For what it's worth, the problem doesn't repro in emacs23 (specifically,
this commit)
commit c7205e5
Author: Aidan Gauland <aidalgol@no8wireless.co.nz>
Date: 2013-02-08 09:44:17 -0500

* doc/misc/eshell.texi: Fill most of the missing sections.

:100644 100644 e05048c... 9fcaa5c... M doc/misc/eshell.texi

On Wed, May 25, 2016 at 12:34 PM Alan Third <alan@idiocy.org> wrote:

> On Wed, May 25, 2016 at 07:18:55PM +0000, Eric Hanchrow wrote:
> > So you're saying that
> >
> > a) Emacs is deferring to Bash to do the completion in this case; and
> > b) bash is not ignoring case?
> >
> > I find both of those hard to believe -- particularly since if I type "cd
> > doc" at a shell prompt that's not inside emacs, but is in a Terminal.app,
> > it does what I want (namely: completes to "cd Documents").
> >
> > If a) is in fact correct, then I suppose Emacs is somehow starting the
> > shell differently than Terminal.app is.
>
> OK, I think I was talking rubbish. It looks like Emacs does do the
> completion.
>
> (BTW, this bug looks like it's OS X specific in debbugs even though
> it's reproducible in a GTK+ build on GNU/Linux too.)
> --
> Alan Third
>