Without being root, is it possible to download a newer version of KDE and use that as your window manager for a RealVNC server?
It's absolutely possible. If you can
sudo, you can install the newer KDE via your favorite package manager (assuming it's available in the repositories).
If you can't
sudo, you won't be able to install anything under /usr/local. You'll need to install to your home directory instead. Anytime the instructions list an "install prefix" variable, change it from
configure scripts are in use, this is easy:
That change will allow you to install your build to your home directory, under
$HOME/share, etc. You'll need to update your login scripts to include your home directory in the PATH and LDPATH in order to access it.
Without root access (be it directly or via sudo(1)), installing something as invasive as a full desktop environment (including matching applets, plugins and whatnot) is probably impossible, and at best very ill advised.
What exactly is the objective of this? If it is just access to a newer KDE, updating the distribution is probably simpler and much less risky.