Updating CB and plugins with package

But if I unzip it I can't install it as a package? Should I install from folder

Just install CB itself from the /extensions/packages folder through Joomla. It's a Joomla package and will install fine there. That should allow the installation error message to output more clearly.

Still only get the same message:

You installed it with the package installer again. You need to unzip the file you downloaded from our downloads page. Next in the /extensions/packages folder you'll find CBs Joomla package. Install that though Joomla extension manager. This should help expose the error, but you should also be seeing whatever the error is in your web hosts error log.

Ok, now I understood.

When I do this the installation never stops. It looks like it's installing and the Joomla label is rotaing, but it never finishes. If I reload my browser after waiting 30 minutes or so, it says "Installation of the package was successful.". If I check the CB configuration it says that I have the new version. So I guess the installation was ok. I don't know why it gets stuck and never shows that the installation is complete.

Sounds like there's a problem with your server then. Check your server error logs or contact your host if you're unsure how and review the error logs around the time of install attempt. Could be something as simple as mod_security triggering a false positive and blocking the install.