Emails stop downloading & some are missed

I am having a problem with POP downloads. I can check the same email with webmail and the emails are there. Some download and others can take hours - even 24 hours. When I boot up in the morning they come down. I can reboot and they come down but another may show up just after a reboot but will not download! I've gone through the various email choices and tried Mailbox - Get new - bellaliant.net and after 10-15 seconds the "no mail ping" and nothing is downloaded and I checked via webmail and it's there.
I've opened /Applications/Utilities/Console and selected the Show All Messages option and watched the output screen while running Apple Mail and it shows...

2014-05-12 11:03:05.373 AM loginwindow[58]: *** WARNING: -[NSImage compositeToPoint: operation:fraction:] is deprecated in MacOSX 10.8 and later. Please use -[NSImage drawAtPoint:fromRect operation:fraction:] instead.

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I'm using OS 10.8.5. Hope you can help...

Here's another clue… I went to Mailbox / synchronize and chose the provider and an email came down instantly. Hopefully that will help someone identify my problem...