There are some publically documented problems with AppleMail due to the way it maintains open connections to e-mail servers. These problems may manifest as authentication issues, or general connectivity problems.
Unfortunately, there is currently no "fix" for these issues until Apple rework the connection issues with AppleMail. Please be advised that these are not server related issues, but e-mail client issues. The only workaround we can advise is the use of another e-mail client altogether.