Someone very early in this thread suggested the issue may be environmental rather than a software problem. I saw the same issue on my iPhone 4s as others have described here.
I run the WU imap server (with no recent changes) working with other imap clients (again with no recent changes) just fine, but not my iPhone after upgrading to iOS 7. I held off upgrading our other phones until this got resolved.
Well, with no resolution on my device II eventually upgraded my family's phones (identical make, model, memory, etc.) and they did not experience the issue! Same device type, same imap server, etc. I noticed my device storage was much closer to being full than their devices. I cleaned things up on my phone to make more storage available. I may have also reconfigured the e-mail client but it has been a while so I do not recall when in the sequence this was done.
So, I'm now in agreement with that early poster suggesting it is an environmental issue - something with the current state of the device - that is causing this behavior. This would explain why it has not been seen in testing and may be difficult (if not impossible) to reproduce by the support team.