Outlook 2013 / BT webmail - inbox displays only latest message and others disappear

I use BT webmail and Outlook 2013 on my PC.  I leave messages on the BT server when I download them. I started having problems yesterday and it got worse today. No one has been able to resolve the issue and I've been told all my settings are as they should be/as expected. Does anyone have any suggestions please?

When I opened Outlook on Thursday, Inbox emails after July 2013 had vanished. New emails downloaded sporadically and then during the day the missing post 2013 emails appeared in batches starting with the older ones first ('synchronising' showed at bottom of screen).  During the evening the post 2013 emails vanished again and then reappeared again in batches. All messages remained on server. Emails on Blackberry were operating normally.

This morning things got much worse. I could see messages coming into the inbox but they would then vanish.  Sometimes 4 or more messages downloaded but within a few minutes I would see only one left in the 'today' section. Messages prior to today could be seen but earlier today I had the same issue as yesterday with older emails. This behaviour (only one message in 'today') was independent of whether I opened or previewed the messages.  The messages simply vanished, and did not appear in any other folder, including deleted items folder. A single message remained.  I have no rules set up.  

Now, as from a few minutes ago, I can now see more than one message in the 'today' tab but not all the messages that should be there, are there, just some of them.

It's terribly frustrating and unpredictable and I don't want to have to use BT webmail unless I have to.  Any ideas what is happening and how to fix it?