You can edit the C:\Program Files\Alwil Software\Avast4\DATA\avast4.ini file, [MailScanner] section and increase the MaxConnections value. You may have to create the line for the MaxConnections value if it isn't there.
[MailScanner]
MaxConnections=20
The default concurrent connection limit is 20 (for mail & news together). If the limit was exceeded, you would see the line Closing connection due to MaxConnections into aswMaiSv.log file. This limit exists not to save resources, but to prevent connection looping in strange configurations, so the user can increase it without worry to, for instance, 50. The value 0 disables the limit.