After moving to a new server I came across an odd issue while browsing mail through IMAP in Thunderbird. Sometimes folder contents were displayed correctly, sometimes not and the whole thing worked in a very sluggish manner ;)
As it turned out, the problem was caused by the IMAP server settings, which allowed only 2 concurrent connections from a single IP. Meanwhile, Thunderbird sets the default number of concurrent IMAP connections to 5. To change it, enter Tools → Account Settings…, expand the branch of the account in question, choose Server Settings and then click the Advanced… button on the right pane. In the new window change the Maximum number of server connections to cache setting appropriately.