I would look into DB connections issues. There is a specific number of DB connections allowed – a setting in the database config (don’t recall which one) – and if there are too many connections, the next connection will fail.
Contact your hosting place and ask them to increase the number of allowable database connections. It’s a setting they have to make, at least at my hosting place; I didn’t have access to the DB config files.
They might want you to upgrade to a dedicated server, rather than a shared server – that resolved a similarissue for me, although I never thought that the site was that busy to need a dedicated server.
My similar (I think) issue was very hard to diagnose, though, so perhaps the ‘you need a dedicated server was just an up-sell.)