I type in the Project URL and my Account Key, and the BIONIC manager shows a project attaching and then detaching, and my message box shows this too.

Please copy and paste some of the messages from stdoutdae, so that we can work out what is wrong.
Wild guess using crystal ball: You don't have enough disk space. If so, go to your account, select the general prefs, and give the project 3-4 gigabytes.

Not this? We'll have to wait for you to re-post.

Your <a href="http://climateapps2.oucs.ox.ac.uk/cpdnboinc/show_host_detail.php?hostid=196467">laptop</a> has been attached to CPDN successfully.

If it's a second box you want to use with CPDN now, you can just copy account_climateprediction.net.xml from the laptop to the new computer.