In working with the hosting company we have already tried a few things. We have three dedicated servers. Front-end, back-end & DB. When we get the error shown above, the same time we get an error on the DB side.
2016-05-27 12:21:55 5949 [Warning] Aborted connection 13527446 to db: 'dbnameexample' user: 'usernamexample' host: '192.168.00.00' (Got timeout reading communication packets).
Our hosting company recommended pushing our connection timeouts from 60s, to 360s. And our max packet size from 16MB to 256MB. These changes helped with our timeout communication packets on the DB. But we are still getting "Can't connect to MySQL server" errors.
We think the next avenue to pursue is to look at why Magento is giving up so soon when the database is clearly online and available. Perhaps this is a php retry or maybe a socket timeout of some kind. We have already pushed the php socket timeouts up to higher values. Any ideas would be greatly appreciated.