To upgrade to a new version, the eclipse FAQ suggests to add a new repository and then install available updates. Upgrading to Neon, the corresponding repository is http://download.eclipse.org/releases/neon/.
On this page, it says:
NOTE: Due to structural changes you cannot update from a Mars (or prior) all-in-one package to a Neon version. If interested in the technical details, see bug 332989 and bug 490515.
I still tried to above procedure, but no updates were found. What is the best way to upgrade from Mars to Neon? A complete new installation? How do I preserve my settings?
The note on the repository page is right. Thanks to your report, I edited the FAQ to make it clear that upgrading packages from Mars to Neon is not supported.
Instead you need to to a fresh install for Neon (or later).