Are you adding IMAP accounts? They can not use local folders.

If you are using POP accounts set the advanced setting in server settings to use the inbox for local folders.

Note that sometimes the folder pane does not refresh properly and you have to restart Thunderbird to get newly added accounts to disappear.

I figured it out! After creating a new account, I would go into Account Settings and reset all the folders to Local Folders, but I was missing the most important one: the Inbox. On the Server Settings page, in order to send incoming mail to the Local Folders Inbox, I must click Advanced in the Message Storage area, and I didn't realize that. Once I do that, and then restart Thunderbird, the account no longer appears in the folder pane.

I understand why some people want to associate each email address with its own boxes, but I used Eudora for 20 years, and Eudora had just one Inbox. I am accustomed to having all my incoming mail going into one Inbox, and that's the way I still want it. My life isn't so compartmentalized that I need separate folders for each account.

I can't say that your comment gave me the answer, but it prompted me to go into Account Settings and examine each page more closely, so you did help me find the answer. Thank you!

Bit by bit I am getting Thunderbird to look and feel very much like Eudora, and that's what I want. I was so fond of Eudora that I used it for a decade after Qualcomm (those jerks) discontinued the program. I kept copying the program files into new directories, but finally they became corrupt. Thunderbird is so much more modern!