Generic Connection Timed Out Browser Error

We host several WordPress websites on a GoDaddy VPS. Sporadically we see a site getting the generic “Site can’t be reached, took too long to respond, err_connection_timed_out” browser error message. It seems to mostly happen when attempting to log into the admin area or when navigating to the admin area for the first time in a day. If this error hits on one of the sites on the VPS, none of the sites on the VPS can be navigated to for what seems to be 5-10 minutes. After that period of time, everything works as expected. It also seems to only happen for a specific person/IP address as it may happen to one of us but someone else on a different IP has no issues. We pay for GoDaddy Managed Support service on the VPS and have talked to them multiple times but they have been unable to point to anything specific. My hope is someone here may have experienced something similar or may have some ideas that can help point us in the right direction. Thanks for any insight you can offer.