OK, It seems to me there is either one setting that is not right.
When you access the mail box that acts as your "server" i.e. ,the on line web site you would normally go to check your mail.....do you see all the folders there that you should see in Mail on your Mac?
Have you set the "send / receive" retreival time to repeat ?
Go to your Home folder (house icon)...select Library...select Mail...select V2...
....select one of the 'IMAP- name" folders and there you should see a list of trays all ending in .mbox.
If you see these then Mail has set them up correctly.
Now what you can do is create online, a GMX account, to see if it is your existing mailbox providers that are at fault. Go to gmx.com and create a new address, with a name you could use. Follow the set up instructions for IMAP and create a new Mail mailbox. Now if this works fine then you have found the problem.. and got one more free email address to boot.
If not, delete the mailboxes you have set up again and remove the 'com.apple.mail.plist' file found in...
...Home folder (house icon)...select Library...select Preferences...scroll down till you see 'com.apple.mail.plist'. select it and drag it to Trash. Then restart Mail as it will automatically create a new plist.
Create the mailboxes again once you have established the info for IMAP client applications (Mail) is 100% pucker.
Do you know anyone else who uses the mail facilty you use? Who are your hosts for the troublesome mail boxes? You could compare the settings.
Also you could ensure that under the Mailbox pull down menu all the Mailboxes have been 'Taken On Line".
Synchronize all the Accounts.
Now I am pretty sure its a setting but if we cant fix it then you can make an appointment with the Genius Bar in all mac stores. I have used them on a couple of occasions and problems solved (Hardware issue) with no cost except getting there.
Let me know how you fare. I am driving down to Brighton and will be back up here on Sunday and will check in to see how things are going.
Michael