Timeout Mail App problem

References backups, you should consider having at least 2 backups on 2 separate drives as hard drives do fail. Time Machine will back up to multiple drives, alternating between them. Another often recommended backup is one that is stored away from your home in case of fire or theft. Since that one isn't backed up daily, I'd recommend a clone. There are also cloud based backup systems you can use for off site storage.

Time Machine Basics

Time Machine Basics (2)

Clone - Carbon Copy Cloner (Often recommended as it has more features than some others)

Clone – Data Backup

Clone – Deja Vu

Clone - SuperDuper

Clone - Synk

Clone Software – 6 Applications Tested

Commonly Used Backup Methods

Cloud Backup Services - Best

Cloud Backup Services - Best (2)

Linc no longer participates here.

Backing up is a rational measure in all cases - especially when beginning any troubleshooting procedures

I also use several IMAP email accounts including Gmail - occasionally they all have server connection problems, though I do not recall "timeout" error message being one of them.

Click on finder take cursor on top menu bar , click on Go > hold option key from down column and click on library .

Select mail folder and right click on the mail itself - click on copy mail .

Right click on the desktop - click on paste item the V 3 folder ( if you are using Mac OSX EL - Capitan ) will appear ( with all of mails in mail application and also server based ) on the desktop , you can drag and drop V3 folder in a usb stick for saving .

This is very helpful particularly for POP accounts also IMAP accounts have the same procedure - as if you delete any mail from IMAP account it is still saved on internet service provider server .

But using POP account if mail is deleted permanently from the server it is gone for ever .

See this article Read Gmail messages on other email clients using POP - Gmail Help

Read these articles also Turn POP and IMAP on and off for users - G Suite Administrator Help

How does Gmail POP work? - G Suite Administrator Help

if it is an IMAP account , the mail data is stored on the servers but it doesn't mean the internet service provider stores mails for ages , it totally depends upon them for how many years they will save the mail data for a user , so for precautions the user must save the data as described above .And deleting the IMAP account is done when you feel a slight glitch is in opening of mails or some major issues are there and never delete mail plist from user library , the very simple method is delete the mail account .

Thanks !

Have you tried going to the Gmail web site and in your account settings, choose allow less secure communications (or wording similar to that)?

Since you've created a backup on the Desktop, deleting and then adding the account back would be the next step.