I just got an iPhone 5S and I am not seeing how to install a me.com email account. I have such an account on my iPad, but the email account information did not transfer over to the iPhone with the rest of my email accounts.
I do NOT use iCloud for syncing or storage and that is the only option I see on the iPhone mail setup that could possibly apply. (me.com loads iCloud.com so I don't even know how I could even change email prefs)
I only can send me.com email by going to the website or using the iPad. The iPhone does not even show me.com in the list of email accounts on the phone.
Fortunately, mail to the me.com address is set up to automatically forward messages to my domain email address.