JurgenFeuchter
It maybe that the user you're trying to use is not having access to that database. We can verify this by opening that database with that user via some other client rather than PHP.
If you have access to the ssh access to the server, you can try this to open the database via command line.
mysql -u USERNAME -pPASSWORD
Then to get the list of databases that user can access type this:
show databases;
If that displays you the database you're trying to use, then it might be the problem you're suggesting, though I'm setting up a dummy server myself just now, installing PHP5.4 and MySQL 5.1 to verify your hypothesis.