We could try repairing your inbox ... some times it helps. In the Folders Pane on the Left find the Inbox folder in your email account. Right Click on the Icon of the Inbox folder (still in the Folders Pane on the Left) and Choose Properties. Click the Repair Folder button then Click OK. (Wait a while for repairs to finish ... ) Now try downloading the messages again. Otherwise you must contact your email provider - the problem is on that end.

Thanks, but this did not help. I tried it already several times. I wrote to my mail provider hoping he can fix the problem, as I have other mailbox with same provider and no issue in downloading to Thunderbird.

There's probably a corrupted message blocking the download of the others.

https://support.mozilla.org/en-US/questions/1158670

Also, empty Spam/Bulk/Junk and Trash through webmail.

If that doesn't help, delete the popstate.dat file:

http://kb.mozillazine.org/Popstate.dat

Deleted all messages in Inbox, Spam, Trash. I deleted popstate.dat file First download worked, but then at next download I am again getting the same error message.

Do you have a security/AV program scanning the mail folders? If so, disable that option by adding the TB profile folder as an exclusion. The AV may be locked on a message and blocking any further downloads.

Thanks for the advice, but still same issue. The only difference since I have deleted popstate.dat file, and all mails in all folders on webmail is that it happens that 1 to 3 message are downloaded before getting the error message.

I disabled the AV mail scanning function totally, tried the download and still getting the error message.

I suggest you give the following a go, including the additional step of trying Thunderbird in safe mode while in the operating systems safe mode.

• Restart Thunderbird with add-ons disabled (Thunderbird Safe Mode). On the Help menu, click on "Restart with Add-ons Disabled". If Thunderbird works like normal, there is an Add-on or Theme interfering with normal operations. You will need to re-enable add-ons one at a time until you locate the offender.
• Restart the operating system in safe mode with Networking. This loads only the very basics needed to start your computer while enabling an Internet connection. Click on your operating system for instructions on how to start in safe mode: Windows 10, Windows 8, Windows 7, Windows Vista, Windows XP, OSX

If safe mode for the operating system fixes the issue, there's other software in your computer that's causing problems. Possibilities include but not limited to: AV scanning, virus/malware, background downloads such as program updates.

Thanks for the suggestion. Restarted PC in safe mode with networking, Thunderbird also in safe mode. But still getting this error message.

Restarted in normal mode, then kept on clicking "Get messages" to download my e-mails, getting the error message, in the bottom bar seeing that trying to download 1 message out of 5. After approx ten attempts of downloads, one message gets downloaded, then again getting the error message, .... so for 5 e-mails it is at least 50 attempts to get them one by one. This behavior is also weird, as if something is blocking, then why after persistence in clicks it lets the e-mail downloading.

Create a new profile and add the account, and make no other changes. This will test if the problem is with your profile or the mail server. Help/Troubleshooting, about:profiles, to create a new profile or run your current profile, which is unaffected by a new one.

I have created another profile under Thunderbird, added my e-mail account and I get same error message during download and same behavior: if I insist, after 8-10 attempts, one e-mail is downloaded.

BTW, I also have another mailbox with same provider (freemail.hu) and no issue in downloading to Thunderbird from that account (of course I have not same volume of messages).

When logging to my e-mail account through webmail from my PC, or my mobile I cannot see the content of all the e-mails, as I got error message. If I try to look at the same e-mail through my mobile BUT using 4G and not my WIFI access, I can see the e-mail content.

Can it be linked to the router provided by my internet provider? Here also no software update was performed before I start to get the issue.

Testing with the new profile and accessing the account by different means certainly suggests the problem lies with wifi access, although there don't seem to be any problems with the other freemail account over wifi. Can you test the problem account with a different wifi network? It doesn't hurt, and often helps, to just reboot the router or restore its factory settings.

Issue still occurring! As it is PC, took some time, but finally I used my mobile to create a hotspot and thus use a different Wifi and by the way a different internet provider also. Did bot help! I also restarted my router, used wifi then cable to connect. Still getting the error message. I also urged my mail provider as still did not get any answer.