Left click the avast 'a' icon, select the Details... >> button if it is displayed, that will show your avast Shields. Select the Internet Mail one and click Terminate answer Yes to the question it won't start on the next boot. This is better than disabling avast completely.
That will allow you to send and receive email, but you have to find the cause of the problem rather than treating the symptom (the error message) by terminating the shield and find the disease (the application taking all the connections). It is also possible to increase the max connections, but that will often mean what was grabbing the connections will just grab more (see below).
What is it that you use minspect.exe for (my google just reveals this, "Remote POP utility - Mail Inspector - iDev.ch ") how do you use it ?
If it is just checking for and reporting the presence of email rather than downloading it, perhaps we can exclude it form being scanned.
Concurrent Connections Error:
You could modify the avast4.ini file using notepad, "C:\Program Files\Alwil Software\Avast4\DATA\avast4.ini." Find the [MailScanner] section and insert a new line after the section heading, and enter this MaxConnections=50 (if that line exists you can change the value to 50). Save the changes, the avast self-defence will ask if it is OK answer yes.
[MailScanner]
MaxConnections=50
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.
