This normally happens because you have told VPOP3 to Leave Messages On Server (on the Mail Connectors → Mail Collectors → (name) → POP3 General tab). If you do this, VPOP3 will leave messages on the ISP's mail server for several days, and remember that it has already downloaded them. Then, the next time it connects it will see that there are some messages waiting, but it may find that it has already downloaded them, so it won't download them again.
See also: Making VPOP3 re-download POP3 messages from the ISP for details on re-downloading old messages.
When VPOP3 connects to an ISP POP3 server it asks the ISP's POP3 server for a list of 'Unique IDs' for the messages. The POP3 standard says that the POP3 server must give each message a unique ID which is different from all other messages and which has to stay the same between download sessions.
VPOP3 compares the list of Unique IDs that are on the ISP's POP3 server now, with a list of Unique IDs for messages which it has previously downloaded, and VPOP3 will only download those messages whose Unique IDs it has not previously downloaded.
Unfortunately, some ISPs use POP3 server software which has bugs in the Unique ID allocation routines, so that several messages may be given the same 'Unique ID'. This is a breach of the POP3 standard, and will cause any software which can leave messages on the ISP's server (such as VPOP3) to believe that it has already downloaded messages which it hasn't.
If you turn on Log POP3 Client Connections on the Settings → Diagnostics → Session Logs page in the VPOP3 settings, and make VPOP3 connect to the ISP, it will write the conversation between VPOP3 and the ISP's POP3 server into a file POP3CLT.LOG in the VPOP3 folder on the hard disk. If you view this file either through the VPOP3 settings or in a text editor like Notepad, you will see VPOP3 send a command called UIDL ('Unique ID List'), and the ISP will return with a list of numbers followed by the Unique IDs for those messages. If you see duplicate Unique IDs in that list, then you should report it to your ISP as a bug with their server, and send them the log file to look at. A decent ISP will be able to see the problem and investigate.
In this case, the only course of action in VPOP3 is to stop it checking the Unique IDs, and just download all messages from the ISP and delete them immediately. To do this, go to the appropriate Mail Connectors → Mail Collectors → (name) → POP3 General page, and turn off Leave Messages on Server and turn off Use Download Rules. This makes VPOP3 fall back to assuming the ISP's Unique ID functionality does not exist.
Also, note that VPOP3 remembers Unique IDs which it has seen but which have been deleted from the ISP in the last few days. This is because, a few ISPs have been known to make messages 'reappear' after they have previously 'disappeared', so this facility in VPOP3 stops those messages being treated as new, and being redownloaded. However, some other ISPs will re-use Unique IDs shortly after the previously using message was deleted. So, VPOP3 may delete the message with unique ID '12345', but then, the ISP gives that unique ID to the next message that comes in. In this case VPOP3 will treat the new messages as old, even though it isn't. To resolve this, go to the Settings → Misc Settings → Advanced settings page and change the Hold Obsolete UIDLs for setting to '0' days. This makes VPOP3 forget Unique IDs once it has deleted them or has seen that something else has deleted them.
There is still the slight chance that re-used unique IDs may cause problems - for instance if VPOP3 downloads a message, then another program deletes that message, and the ISP gives the same unique ID to a new message before VPOP3 reconnects. VPOP3 won't know that the message with that unique ID has changed, so it will treat it as old. In this case, the only solutions are to ask the ISP to change their POP3 server configuration so it won't re-use unique IDs, or at least waits a reasonable time (at least several days) before doing so, or make sure that no other software access the ISP's POP3 mailbox other than VPOP3.
Also, making this setting change may cause duplicate messages to be downloaded if, for instance, the ISP doesn't actually delete a message after VPOP3 has asked it to.
(See also VPOP3 Deleting Messages as Old Messages )