I don't think cookie is related. This cookie helps driving "debug session" -- meaning if you debugged one page and then clicked url or somehow else navigated to next page on the same domain, it would fire or not fire debug session. DebugBreak() is kind of trigger debugger now. If it failed, the reason can be -- it didn't know how to find the IDE's IP. You can try to set IP in the DebugBreak parameter. For example if it is 1.2.3.4, it can be called like DebugBreak("1@1.2.3.4;d=1"); The IP address is 127.0.0.1 in case of local debugging. Also it is 127.0.0.1 if sever is remote and you're using SSH tunnel straight to the server host.
If you have "debugger.fail_silently" Off in debugger section on the server, you may expect failures logged in the php log stream -- this may be a better way to chase the problem.
In any case - make sure you're running the latest debugger module. Currently it is 11.0.23, see
http://www.nusphere.com/news/changes200.htm