Reinstalled Mountain Lion.
Created a new log in, could read the emails for the icloud account I used for the log in. Could send emails. Tried to add my other three accounts it froze on the first one. Waited for about 20 minutes, it moved to step two of created a new mail account. Several more long waits until got to final step (probably over an hour total), waited so long fell asleep. Woke up next day and second account was added! But didn't work. Back to square one (four account total so could be a long time).