The normal reason for message downloads not completing is that a third-party antivirus program is scanning the messages and getting stuck at some point.
We recommend that the first thing you do is to temporarily disable the email scanning part of any antivirus program or Internet Security software on the VPOP3 computer (other than those integrated fully with VPOP3 - i.e. 'avast! for VPOP3' or 'Sophos SAV Interface'). Usually that will solve the problem.
Also note that old versions of the 'MailScan for VPOP3' software would occasionally lock up, so check that you haven't got an expired version of that installed on the VPOP3 computer. Up-to-date versions of MailScan should be OK.
If disabling the antivirus/Internet security software does help, then you need to contact the technical support for that software to help further.
If it doesn't help, then go to Settings → Diagnostics in the VPOP3 settings, turn on Log POP3 Client Connections and set Log Level to Full Logging. Then, when the problem occurs, send us full details of the problem along with the VPOP3.LOG, VPOP3.LBK, POP3CLT.LOG and POP3CLT.LBK log files to firstname.lastname@example.org.