Are you using a secure mail key instead of the account password?

https://support.mozilla.org/en-US/questions/1330660

Thanks for your answer. I created secure mail key for this account (took me some time to figure out how to do it), but now this account is OK with sending and receiving e-mails. For some reason the other account is still using old password and everything seems to be working.

The other account might work for now, but if its password is changed or it's added to another device with a desktop mail app (as opposed to a phone or tablet), apply a key.

Have one more question. On my main pc everything is working for this account. On the other PC I had to enter the same password for inbound and out bound servers, but when I closed and opened Thunderbird it didn't remember it. When I look in saved logins I don't have an entries (inbound and outbound) for this account. How can I add them? (It saved on my main pc)

When you're prompted for a password (and you enter the secure key for AT&T accounts), there should be a box to check to have the password (key) saved in Saved Passwords. It can happen that the checkbox is missing, or there's an external password manager intercepting password entries, such as provided by some security software.

http://kb.mozillazine.org/Password_not_remembered_-_Thunderbird

I saved inbound password in the password manager, but I had to restart the client several times (it was asking me to enter the password again). Finally after 4th time it stopped asking for password. but I still don't see an entry for inbound server in the save passwords, is this normal?

It's not normal, but you have to answer the question about the checkbox and external managers, if any. The kb article explains how to delete 3 files if passwords aren't remembered.