Why can't I get emails on my iPhone 5?

How can we provide suggestions? You've provided no details about your problem whatsoever.

If your getting email on your iPad, you should be able to get them on your iPhone 5. You could delete your Mail Account on the iPhone and just start again by adding it back. This process typically only requires selecting which email provider you use as you Add Mail Account. Then you just need your email Account ID and your email Password. Should work.

Have you looked to see if your iPhone has any Restrictions set that would Block the email?

Does apear any failure notice? Do you use imap or pop3. If one account cannot connect at the same time like the other one it could be a restriction from provider that just accept one connection, so you can load the other account after disconnecting the first account. in this case you get a notice that your username or password is wrong.

Great idea, except that it does not work. I am unable to add my second gmail account to the iphone. It recognizes the first one, actually even recognizes the second (and tells me that it has been added to the phone), but when I go to MANAGE ACCOUNTS, the second account does not show. When I try to add it, it seems to add it, but then it throws me back into the first mail account and leaves the second one empty - totally ignoring my request to add it. I don't know if this makes sense... but it is super frustrating as I cannot get email from my second gmail account at all.

OOk thats maybe a failure of autoconfiguration of iOS. if you fill in your mail address it looks for the right settings. You could try to fill in a wrong e-mail with no hoster ending e.g. gmx.com . So it won't find setting and you can set it manually. Maybe know you can save it as separate account.

GGo to setting > mail and press add new account. Then fill in a false e-mail like example@example.com so iOS cant find configuration for that hoster. At next step you can fill in your right e-mail, password and mail server.