I know this is an old thread, but for those that follow....
For an answer to this, try (long version):
http://premium.wpmudev.org/blog/warning-your-iphone-or-mac-mail-may-be-killing-y our-website/
or (short version):
http://support.hostgator.com/articles/specialized-help/email/setting-up-email-fo r-the-apple-iphone
Too Many Connections - 500 Errors?
On the iPhone there is a function called "push," and when it is enabled it keeps the connection to our server open and continually checks mail. Consequently, this creates many IMAP processes and eventually conflicts with our Terms of Service.
To avoid running excessive IMAP processes, you can disable push when you don't need it.
- Go to Settings > Mail, Contacts, Calendars > Fetch New Data.
- Toggle Push to Off.
Now, instead of retrieving emails continuously as they arrive, your iPhone will use the global Fetch setting to retrieve emails manually or at intervals.
This solved my equivalent issue with Hostgator and my wife's iPhone...