When new email stops arriving at your Droid or you see only very old emails the solution was to remove the email account and then to add it back again on the Droid cell phone.
There is a trick to delete the account.
From The Email listings press menu button next to the House at the bottom of your droid
Select Settings then Select Accounts
You will see your account and the heading Add Account. Yo need to bring a menu to delete the account which is done by pressing and holding your account name. It is NOT enough just to tap on the account name. Then you can delete the account. After that you may add the account in the normal way with Add Account.
You may also need to check the Frequency when email is fetched ( it says automatic push on mine) and the number of messages to retrieve. I set mine to 200.