Short answer

It's not possible to use a Google account as alternate email address of another Google account.

Brief Explanation

It's very likely that the referred account is still associated to a Google managed account even if you don't have access to the Google services with that account anymore and the emails are being forwarded to your Gmail consumer account.

It's worth to mention that the domain admin could set email routing rules to automatically forward emails to an external email address or another email service and and keep the account in the domain user list while blocking the access for that account to the Google managed services.

Extended Explanation

In 2006, Google announced that domain owners could use Gmail as their email service. Over the years it evolved to Google for Work, a brand that encompases several Google product and services including managed Google accounts, which formerly was referred as Google Apps accounts. In the last years the Google Apps term is used broadly used even by the Google employees which lead people to confusion besides that Gmail could be or not enabled for the Google managed accounts.

To check if an email address is the username of a Google managed account

1. If you know the account password, go to www.google.com/a/yourdomain.com and sign in using the email address and password.
2. If you don't know the password search the email address in services like Google+, Google Contacts, Google Drive (sharing a file with the email address), etc.

To check if a domain have managed accounts go to

• sites.google.com/a/yourdomain.com
• groups.google.com/a/yourdomain.com

If the domain doesn't have any managed account you will get a "Domain not found" error, otherwise, you could get a message telling that it's required an authorized account or to get access to the home page of that services.

If the account is a managed account but it is suspended, the user could not access to Google services using that account.

References

• Big mail on campus - Official Google Blog

The instructions at http://www.tekgrl.com/accepting-google-calendar-invites-to-a-non-gmail-account/ worked for me, taking into the account the comments at the end. Here's a summary:

Google has finally fixed this, all you have to do is enable a setting in your Google Calendar to be able to respond to any calendar invite sent to any of your Gmail alternate addresses. Here’s how you do it:

1. [from the comments] Set up the alternate email address a@example.com for your Google account. The option in Google Calendar below may not show up unless you have. Here is Google’s help article for setting up alternate addresses.
2. [from the comments] Wait some necessary time for the Google Calendar to recognize you've added the Alternate Email from step above.
3. In Google Calendar (on a@gmail.com account`) click the gear icon and choose Settings.
4. Select your primary calendar in the Calendars Configuration menu item on the left pane.
5. In the “General Notifications” section for your primary calendar, check the box “Allow me to respond to event invitations forwarded from these addresses.”

Note that invitations already sent before you set this up won't work. Only new invitations will work properly.