Yes, I can do that.
I have an Xbox Live Account, an Xbox Live Gamertag, a Windows Live ID, and a Windows Live Password.
I log into the Xbox Live Account from the PC using my Windows Live ID and Windows Live Password and am able to see the Xbox Live Gamertag on my Xbox Live Account.
I then get on my Xbox 360 and click my Xbox Profile.
I then click "Join Xbox Live". It then asks for a "First Name".
I shorten my Xbox Live Gamertag and enter it in the First Name area.
It then asks me for my Windows Live ID. I enter my Windows Live ID.
It then asks me for my Windows Live Password. I enter my Windows Live Password.
It then tells me that the Xbox Live Gamertag I entered is not associated with the Windows Live ID I entered, and that I should enter a different Windows Live ID and Windows Live Password.
Thus, Xbox is confused.
I know the Windows Live Password I entered is correct, because I can log on with it on the PC.
I know the Windows Live ID is associated with the Xbox Live Gamertag because they're both connected on my Xbox Live Account.
It's very odd, yet somehow funny, to be called Wobbly...