Due to a screen issue, I had my iPhone 5 replaced two days ago. I repeated deleted the mail account and kept adding it but this did not make any difference. When I attempted to setup the new phone with my Google (Gmail via exchange active sync) accounts I was receiving this same "Cannot Get Mail" error. From what I was able to gather, this problem should be affecting Gmail accounts set up via Exchange. When originally purchasing the phone I was assisted and directed to set up my Gmail account through Exchange because it had the ability to use "ActiveSync" and push not only mail and calendars, but all of my contacts as well. I found this article regarding the subject.
http://www.pcmag.com/article2/0,2817,2413283,00.asp
What it came down to was having to delete my Gmail via Exchange account and separate it into two different accounts, a regular gmail account for my mail and a CardDAV account for my contacts.
Here is what I did. Google is changing their system and for new registrars Exchange ActiveSync will not work so you will be unable to register your account via Exchange and have your mail, calenders, and contacts all pushed together. I find this change really annoying. If you are having problems accessing your contacts after deleting the mail account, like I was, here is the work around.
To circumvent this issue go to Setting > Mail, Contacts, Calendars > Add Account > Other > Add CardDAV Account: Under server ender google.com, then your email user name and password, then click next to verify the information.
I hope this helps.