Once you upgrade to 1.7.1 you have a 1.7.1 database. You cannot got back to 1.6 without restoring the database with the back-up that you made before the upgrade.

Allright I thought so... But do you have any idea, why the new Version can't access the Elgg Data files? The location is the same and I haven't changed anything concerning the read and write permissions.

The upgrade from Elgg 1.6 to 1.7 changes the directory structure from using usernames to using the date that people joined and their unique id. This makes it easier to support usernames that use special characters or national characters. The upgrade should have moved all the files and directories over to the new structure. If that didn't happen there may have been a permissions issue. It's hard to comment without more information.

In the old directory structure there would have been directories like p/k/u/pkurzok and in the new one it might look like 2010/07/27/897/

If Elgg can write to the data directory, it should have been able to create those new directories and move the files.