It is so strange to me that everybody is getting different responses from Apple.
I called yesterday to check on my case number and the advisor I spoke with acknowledged that Apple is having a push issue with many of its iCloud accounts, and that the issue is believed to have been caused by massive surges in server usage upon the launch of iOS7.
This comports roughly with my experience, seeing as how I was running a 4s with iOS6, and my push email stopped working right around the launch of iOS7. I am now running a 5s (on iOS7, obviously) but my push email for my iCloud account still does not work.
I haven't escalated my claim to a Senior Advisor yet because every time I have Apple Support call me, the advisor on the other line is already fully aware of my problem and assures me not only that Apple is coming up with a fix but - more importantly - that Apple is aware of this specific issue in the first place.
For what it's worth, on my call yesterday I was told that a fix would likely come in an update to iOS7. This isn't very reassuring as that I'm fairly certain the issue is on Apple's servers, but seeing as how push seems to work fine on my Macbook Pro, maybe it is an iOS issue. There are some issues with iMessage that are getting a lot more attention right now, those are probably at the forefront of their attention.
A note to all of you: When you do speak to Apple, please try and be very specific about what your problem is. I generally explain away any potential fixes that they may offer me before they are even able to by expalining that we have all tried restarting, restoring, re-setting up the iCloud account, etc. etc. What's better is to just politely tell them that you are willing to hold a moment while they check their logs to see if there is any internal report circulating at the moment regarding a persistent issue with iCloud accounts not pushing e-mail to devices. This will help focus Apple's attention on the fact that this is a widespread issue, and perhaps will allow us to "band together" so to speak in getting our collective issue addressed.