Does not appear to be the case, "Guccifer" was the first to bring this to light after he broke into Sidney Blumenthal's AOL account, they came from "hdr22@clintonemail.com".
Hah! Try spoofing an email to a recipient in the State Dept and tell me how it works out :)
Find me someone in the State Department willing to be the recipient and I will happily conduct this experiment. Why do you think it would be any harder to spoof an email to the State Dept than anywhere else?