I've had this problem for at least 2 months. I also have applecare. They moved it up the chain and the senior advisor-1 wanted access to my email account to look into it. I was hesitant but figured I'd do it for a day or two. After 4 days or so I emailed back and forth with advisor in charge asking how long they would need access because I wasn't comfortable giving them indefinite access. He said he had no idea and it would take as long as they needed. I found that unacceptable and changed my password so they couldn't get into the account anymore.
Almost three weeks later the same advisor emailed me again asking if I had changed my password because they couldn't get into my email account. So it took them 3 weeks to even begin to look into the problem??
I called another senior advisor-2 and told him my original alias problem, and the issue I had with their solution to it (unlimited access). He said that didn't sound right and that he'd look into it. He called back and told me that HIS email account was having the exact same problem and that another senior advisor would contact me. He (sr.advisor-3) did and requested access to my email account AGAIN. I called the advisor-2 who was having the same problem that I was and asked him why they couldn't experiment with his email account instead of mine. His (sr.advisor-2) reply was as lame as the proposed solution, he said because I'm the one who put the help ticket in for it. Seriously? Once again I denied them access to it.
So as of right now I still can't create any email aliases. I've only got one and it says I have 2 of 3 available. My guess is they have no idea how to fix it and are ignoring this issue.
My solution: create another gmail address. Since Apple insists that we use an outside email address for our Apple ID I have a gmail account anyway. Gmail doesn't seem to care how many email address I want so I'm sticking with them.
Has anyone found a resolution to this yet?