Update:
I just got off of the phone with an Apple engineer (yup, and it really wasn't that hard to insist on speaking with them). Here's the scoop...
Apple identified months ago that Cox email servers do not have the "Idle server command" option available and/or initiated. This causes the inconstancies that are becoming more of a problem now with Apple Mail in Mac OS 10.8x and 10.9x. This has been reported to Cox multiple times, yet no changes have been made. Until Cox corrects this issue, more and more customers who use Macs will experience the same frustrations.
Cox, please show the same level of customer service attentiveness as Apple and correct this issue for your customers!