I have been using Xperia U on ICS and never faced this issue. With my 1 week old Nexus 5X I sporadically get notifications for my gmail app i.e. sometimes (10%) it shows notifications but most of the times it doesn't. This also happens even if I am using my phone. So definitely nothing to do with deep sleep. I manage my personal gmail account as well as university email. Please tell me some solution to this problem as I am missing important mails from my university and am considering returning this crappy phone.