postmaster says cant deliver sent email

We are having a problem with our email being sent out wither a single or a group. The message comes back from the *** Email address is removed for privacy *** that our mail wasn't delivered. This only started when our accts Hotmail & live  were changed to outlook & stopped linking the accts together so that we could jump from one to the other. We have spent way to much time trying to correct this problem without any success.