Can that account be set up as IMAP? You could try to do that and see whether that sheds some light on what could be the problem…

Can that account be set up as IMAP?

in principle, yes. But I have 10+ years of email history locally in my thunderbird profile and only exactly 1 year on gmx.net (due to storage limitations) and I am afraid when I change that to IMAP, all emails older than 1 year will be gone...

I'm not asking you to change or remove your current POP account, but rather to set it up as IMAP while still keeping the POP setup you have (you may have both setups alongside each other), just so you may (1) check whether IMAP works and (2) be able to see what does Thunderbird see on the server.

And storing all that mail in your POP account Inbox, if that's what you're doing, is a bad idea anyway, could be related to the problem even. You may create folders under Local Folders, at the bottom of the list of accounts in the folder pane, to store and organise your mail locally on your computer there however you wish, independently of any account you may have.

[...] set it up as IMAP while still keeping the POP setup you have (you may have both setups alongside each other), just so you may (1) check whether IMAP works and (2) be able to see what does Thunderbird see on the server.

This does indeed work

And storing all that mail in your POP account Inbox, if that's what you're doing, is a bad idea anyway, could be related to the problem even. You may create folders under Local Folders, at the bottom of the list of accounts in the folder pane, to store and organise your mail locally on your computer there however you wish, independently of any account you may have.

Hm, interesting point. I do have some ~20k emails in that inbox, size is 2 GB in total of the Inbox folder, but I also have a bunch of folders under that account into which I sort emails (i.e. they are on the same level as the Inbox folder, and they of course only exist in thunderbird). Would that be any different than having the folders in "Local folders" ? And is 20k emails too much, and can I somehow fix the not-working connection of my pop3-account by "simply" cleaning up the "Inbox" folder?

Thanks!

Creating folders within the account makes more sense when it's an IMAP account, because in that case you may want your mail to be organised into folders at the server. But for mail stored locally, the use of Local Folders would be preferable, because mail accounts may come and go, you may want to have several of them, you may want to delete one that's having problems and set it up again, etc. Mail stored within Local Folders isn't affected by any of that.

I'm not aware of any hard limit on the size of a folder or the number of messages within it, but the more messages you store there and the bigger it becomes, the more likely it is that it may become corrupt and that you may eventually lose access to the mail stored there… to all the mail stored there…

This is because the contents of each folder are stored in a single text file with one message after another in mbox format. Things like software bugs, hardware failures, or malformed messages, may eventually cause Thunderbird to lose track of where a message ends and the next starts, and this is more likely to happen precisely with Inbox, because of the high number of message additions and deletions that happen there, so keeping that many messages stored in Inbox is definitely a really bad idea.

I don't know whether this could be the cause of the problem you have, I'd expect Thunderbird to report an error message if it was failing to add new messages to Inbox, but it's certainly a possibility to consider.

So yes, I would recommend you start by moving all the mail you currently have in Inbox into other folders, and then, when empty, right click on it and choose Compact. Don't compact the folder until all your mail has been moved out of there, because if there is some mbox file corruption, compacting the folder can only make things worse. Compacting is good maintenance practice when things work well to help prevent corruption, but shouldn't be used if there is some corruption already:

https://support.mozilla.org/kb/compacting-folders

I don't really expect this to fix the problem, but is something you better do anyway before trying other things.