Q: iCloud iMap connection problem

So I bought a new (early 2015 retina ) MBP set up mail and am synching to all devices. I have checked all settings, I have deleted and added the accounts but I continue having a problem connecting to my roadrunner and iCloud iMap servers. When checking connection doctor smtp is green on both accounts but cannot seem to connect to the iMap servers on either account.  All works as it should on my iMac (same network, same settings, same everything, I checked!), on my iPad, on my iPhone.  I have this problem ONLY on the MPB. The moment I check my mail or attempt anything I get the "account error" message.

and the account goes off-line.  If I quit mail and restart it, it'll work for a while till it goes back to "offline" with an account error.

Now, I do have problems every once in a while with roadrunner.com (TimeWarner) and I can usually wait those out.  But as a rule I have no problems with iCloud imap.mail.me.com.

Any help appreciated.

Helga