When the tree view has keyboard focus, you can't drag the inspector. You have to click once in its title bar,
which makes the tree view lose focus, and only then you can drag it.
This is caused by the tree list being a NSView.
This is fixed by my WebInspector branch.
*** Bug 13334 has been marked as a duplicate of this bug. ***
This was fixed on the inspector branch and merged to trunk in r20986.