UPDATE FROM MY ORIGINAL POST***
You do not need to delete your email account from your phone.
I've posted this on other similar threads.
How I fix this Issue.
I have 6 different emails set up on my Droid Max and I had the same issue with my Droid X. Periodically for no reason my email accounts will start giving me messages that it can't log in. It may happen to one or two of them, or all six of them will suddenly not want to log in. Sometimes, if I ignore it, it resolves itself but I can't go 24 hours without being able to get to my emails via my phone. It's a real PIA.
My experience is, this has nothing at all to do with the settings on your phone. How could it be, I've set up my accounts and used them for months and months with no problem and then boom all of the sudden it can't login? My conclusion is that this is caused by the security utility from Yahoo and that it periodically gets hung. May not be the solution for everyone but it always works for me.
Here is what I found that has worked for me:
Log onto your Yahoo account. From another device, like a Laptop.
There are several ways to do this.
I usually just use my laptop and log into the Yahoo email for whichever account is giving me issues.
Go to the "gear" icon in the upper right hand corner and select "account info"
It's currently the last selection on that list. It will make you log into your account.
Go to the "Account Security" menu option on the left.
On the bottom of this screen is an option for "Allow Apps that Use Less Security Sign In"
Mine is always set to "On",
click on the switch icon to Turn this "OFF"
Now, click on the "Preferences" menu option, let that screen load.
click on the "Recent Activity" menu option, let that screen load
click back on the "Account Security" menu option
Turn that same option "Allow Apps that Use Less Security Sign In" back to "On" by clicking the switch Icon again
Now close the "Account" screen and go back to your email screen. You should now see two emails from Yahoo stating your Account Options have changed. Once I see these two emails, I go back to my phone and it is usually working again and I can see those two emails from Yahoo stating your security settings have changed came through on my phone.