How do you have the same account setup twice on Tbird? Which protocol is each account using, IMAP or POP?

The one which keeps the e-mails is a POP account. The one which loses e-mails is an IMAP account.

Whenever I start Thunderbird I get the following error message from the POP account:

"Unable to connect to your IMAP server. You may have exceeded the maximum number of connections to this server. If so use the advanced IMAP Server Settings dialog to reduce the number of cached connec ..."

The POP account is responsible. Unlike IMAP, POP typically fetches messages and doesn't leave a copy in the mail server from which the message was downloaded, unless you specifically configure the client (Tbird) to leave a copy of the message on the server. You need to do so on each device where you've configured the POP account. If you don't do this, the device that fetches the email first will not leave the server's copy, meaning the same message won't be accessible from a different device/account setup. You need to open Thunderbird's Account Settings, select Server Settings for the POP account then tick the checkbox that says leave messages on server and below that select how long you want it to keep the server's copy.

Thanks for the reply. I do not want to leave a copy on the server, I just want to download and keep the e-mails. The trouble is that the POP account comes with the error message whenever I start Thunderbird.

In that case, you should know that IMAP gives you a remote view of what the server contains in real-time, so you will not see the messages that have been fetched by the POP account setup. What error message exactly do you get and who is your mail provider?

My e-mail provider is Ziggo and the message I get is:

"Unable to connect to your IMAP server. You may have exceeded the maximum number of connections to this server. If so use the advanced IMAP Server Settings dialog to reduce the number of cached connec ..."

Correction: The message I now get is:

"Unable to establish TLS connection to POP3 server. This server may be down or may be incorrectly configured. Please verify the correct configuration in the Server Settings for your mail server in the Account Settings window and try again."

Check those settings and make sure they are same as these https://www.ziggo.nl/klantenservice/internet/e-mail/serverinstellingen

Excellent! I had already contacted Ziggo and requested this information but they were unable to provide it to me. Nor could I find it on their Website. I am glad that you found it. Well done!

Found it via a Google search for "ziggo mail settings". You're welcome.