If you used an early version of the upgrade files for 1.5.16 (not the full release) you may have incorrect file permissions of 775 instead of 664 for your files. This was do to an error in the automatic packaging of those files.
To resolve, the simplest solution is to overwrite your installation with the full package.
How to do this will depend on how you usually install, but for example:
Upload the full package to your Joomla! Root.
Extract the files (if you are asked about overwriting existing files say yes).
DELETE THE INSTALLATION FOLDER.
As always, if you have made any changes to core files, back them up before starting.