Mac email doesn't work with iCloud

Question

Since the move from mobileme email to iCloud I can only get email accounts working for a short period, and sometimes not at all on agiven day.
What we had on mobileme was a family acount with 4 active email accounts.
These all worked fine and all came in as seperate mailboxes to the mail client on the Mac. Nice and exactly what we wanted.
Since iCliud these accounts all work fine on IOS devices where the account matches the appleID logged in on the account.
Email on the Mac is different. I am often asked for the password for the individual accounts registered within email. Sometimes these passwords are accepted and I can send and receive email, sometimes they are not. Neither of these things sound right to me a) why only ask sometimes? b) if it works once why doesn't it always work.
I have tried (over 20 times now - waiting for varying lengths of time up to several days) the following to find a solution:
1. Finding a post on these support forums or elswhere that says it has the definitive way of getting the whole account thing sorted out. It does seem many people have had issues moving email to iCloud.
2. Creating new accounts for the email addresses required following the instructions exactly and having someone else check that I am.
3. Trying the account immediately (actually email just does this). Sometimes it works sometimes it doesn't
4. Waiting  and trying it later. Sometimes the account will work for a couple of days but eventually everything reverts to the asking for and rejecting/accepting the password.
5. Going back to step 1
Other iCloud services work fine e.g. Calendar synch.
Any help gratefully received. This has been going on for months.

Answer 1

I'm sure you migrated the primary MobileMe account to iCloud, but did you migrate each of the other email accounts as well.  Just migrating the primary account does not migrate the others.