>  Are there any other versions of libpng16-16.dll on that system?
>
> I ran the following:
> $ find.exe / -name libpng16-16.dll
> /cygdrive/c/Portable Programs/emacs-26.0.91-x86_64/bin/libpng16-16.dll
> /cygdrive/c/Portable Programs/inkscape/libpng16-16.dll
> /cygdrive/c/Program Files/emacs-25.1-x86_64-w64-mingw32/bin/libpng16-16.dll
>
> I found the old C:\Program Files\emacs-25.1-x86_64-w64-mingw32\bin on my path, removed it and restarted
> Emacs, verified that it was not listed in (getenv "PATH"), but no luck. I still cannot display PNG images.
>
>  Does pngfix.exe work (it should be in the same directory as
>  libpng16-16.dll)?
>
> Yes it does.
>
> $ ./pngfix ../lib/python2.7/idlelib/Icons/idle_16.png
> IDAT OK  maximum 12 12 1070 2064 ../lib/python2.7/idlelib/Icons/idle_16.png
>
>  And finally, what is the value of libpng-version in your Emacs?
>
> libpng-version
> 10634

Strange.  I can only suggest stepping with a debugger through
init_png_functions, to see what exactly fails there and why.