Mail App keeps asking for my email passwords on Mac OS-X Lion

I've had this problem with every single MacBook I've owned running on Leopard, Snow Leopard and now on my Lion one too!

Does anyone know how to stop it? It seems to be worse than ever on my new Lion OS MacBook Air.

I have Yahoo!, iCloud, Hotmail and BT Internet email addresses in my mail app. Every two minutes the dock icon starts bouncing and it's requesting the password for one of them. Everytime I tick the box saying "remember this in my keychain" but no sooner have I done that and it pops back up again requesting the password that I have just entered. I am entering the correct passwords and all email addresses are active.

It's really winding me up at the moment as I keep having to quit the mail app because I can't stop it from asking for passwords. 😠 This MacBook is brand new. I only got it delivered from Apple last week!

If anyone can shed some light on this, I would be very grateful!

Thank you. 🙂