I think I have a fix!
So just to add a slightly different spin on this thread, I thought I was having exactly the same issue, but I have just discovered that SOME of my emails are in fact getting through to the receipient, but I just can't see them. It is like they have completely vanished. It is not showing in my sent box on my iPhone (iOS 7.0.4), or in my Outlook sent box, or even if I log into webmail it isn't showing their either.... but the email was received by the person I sent it to!
Some were just completely disappearing. Not being recieved and not showing up in my sent folder on my iPhone, Outlook or Webmail. I have just done some tweaking and it seems to have worked. This is what I did:
Settings> Mail, Contacts, Calendars> (choose the mail account that is giving you grief)> Account> scroll down and select "Advanced".
You then have a screen that allows you to select "Mailbox Behaviour". Choose "Sent Mailbox". Mine was set to the "Sent" mailbox under the section "on my phone". Change it to the "sent" box under the section "On The Server".
I also did the same for Drafts and Deleted mailbox. This seems to have worked. I have run a few email tests and it's holding together for now. Hopefully it sticks and I hope it was this easy. I was just about to do a complete re-install, which I was dreading.
Here is a screen shot of the screen that I changed.
