With "Allow multiple copies of Toad to be loaded" checked, I still can't open another instance of Toad. When I click on the Toad shortcut icon again, it closes my existing instance of Toad and re-opens it, causing me to loose my work.
It is possible for IT Admins to restrict how many sessions of an application you can open.
Please check with your IT Admin to see if there are any policies to restrict certain applications from running more than one instance.