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I have the same problem, iphone 5 and IOS 7, although the emails I sent yesterday did turn up in the sent folder eventually (when I checked this morning)

Strange.... I have PUSH enabled for the account and it is a .me account, is your account a iCloud account?
 
Not iCloud. It's for a buddy. His settings show a pop account using his company domain. Can't figure it out.
 
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