How many messages does your Inbox folder contain? Any error message when Thunderbird tries to download emails? Have a look on the error console via Help > Developer Tools > Error Console

How many messages does my Inbox folder contain? It says 4 new but the are not shown. Any error message when Thunderbird tries to download emails? Yes, it says I need to delete some message...I did that but new ones still wont download.

I guess that the warning you get is "There is not enough disk space to download new messages". R-click on the Inbox folder > Properties > General : which is the number of messages and which is the size on disk?