If your alumni email account exceeds the 2GB quota, outbound email (email you send) will fail to deliver.  If you go over 4GB, both outbound and inbound (email you receive) will fail.

You can check your storage usage by going to your Google account storage page.

To reduce storage, see Search operators you can use with Gmail to filter and search for emails that are no longer needed (e.g., that may have attachments or older emails before ‘x’ date).

It’s a good idea to delete emails when no longer needed or they have been forwarded to a personal account. 

Consider setting email forwarding from your alumni account to another preferred email account. 

You can “send-as” your alumni address from your preferred email.