Summary

Some emails are coming in but are not being processed into tickets.

Environment

Gmail Email Server

Diagnosis

The Too many simultaneous connections via protocol imaps error shows up in the mail logs.
There is a significant number of mail handlers all connecting to one Gmail account.
The error means that the mail host (gmail in this case) is limiting the number of concurrent connections from any single source.

Cause

The error means that the mail host (gmail in this case) is limiting the number of concurrent connections from any single source.

According to Google documentation, the maximum number of simultaneous connections to a gmail mailbox is 15.

You can only use 15 IMAP connections per account. If you have many email clients getting messages from Gmail at the same time, you may see the "Too many simultaneous connections" error.

Possible Solutions

Increase Mail Handler Delay

A mail handler has a delay setting which dictates how often it will poll the mail server for new messages. Increasing this delay can help to reduce the number of connections that Jira makes to Gmail. You can edit this setting in the mail handlers configuration.
In previous instances of this error, setting the delay to 5 or more minutes has helped to reduce the number of simultaneous connections to the mail service.

Larger intervals

Space the mail handlers further apart to reduce the possibility of mail handlers running at the same interval. This won't have a guarantee to fix the issue, but it would lessen the occurrence of this error.

Split emails into two or more accounts

The limit that Gmail imposes on IMAP connections is set per account.
A possible workaround is to setup separate email addresses for some of the mail handlers. In effect it would have lesser mail handlers competing for the IMAP connection limit.
This way, the mail handlers that are working on a different gmail account won't count against the 15 IMAP connection limit of the other set of mail handlers.

Switch some of the mail handlers to POP

These errors are due to IMAP connection limits. If some of the other mail handlers are switched to POP protocol, they won't count towards the 15 IMAP connection limit.
Please do note that Gmail POP has its own set of limits as well. The Gmail documentation says that POP clients accessing Gmail account too frequently may run into problems too, but there seems to be no exact number mentioned.

If you see the "Account exceeded POP command or bandwidth limits error," it's usually because your POP clients have been accessing your Gmail account too frequently.

To fix this, change your client's settings so it won't check for new messages too frequently.