After reviewing Google's Security Settings, Mozilla Thunderbird could no longer connect to Google's servers and retrieve any e-mails. It said that I use several less secure apps, and therefore access to gmail servers has suddenly been blocked. Nice!
I mean it's so nice to see how Google keeps working on security, but usually that also means, that one has to adjust app settings in e-mail clients to be able to get to the e-mails, stored in gmail.
The Solution for Access Blocked
In Mozilla Thunderbird I had to change the Account Settings/Server Settings and change the Authentication Method from 'Normal Password' to 'Oauth2', saving new settings and reentering my password to sign into the gmail account.
Allow Access of Less Secure Apps Again
Click here, log into your gmail account, and change the setting.
It could happen that quite a few, if not all, non-Google e-mail clients on Android stop working. At least HTC Mail, which I used most of the time, doesn't fetch e-mails any longer, in the end it even lost the account settings.