My e-mail account is SKY, but my preferred e-mail provider is Windows Live Mail, when I open my laptop in the morning, I may have 6 e-mails in Sky, but nothing in Live Mail, however once the laptop is running I receive e-mails that are sent whilst the laptop is open but not the ones that are sent when it is switched off.
As you're accessing your emails via Sky first, I think what you may be seeing is Sky removing the emails from the server once they're downloaded. Thus when you later try to access them via Live Mail, there's nothing left in your server's inbox. When you start your PC in the morning go directly to Live Mail first and your mail should be there.
You're still getting them in Live Mail throughout the day because Sky hasn't downloaded or marked them as 'read' yet.