I have 4 VMware ESXi 5.1 hosts and 2 vCenter Servers

vCenter Server #1 manages Host A and Host B
vCenter Server #2 manages Host C and Host D

Starting about a month ago... when I launch the vSphere Client and connect to vCenter Server #1, after just a few seconds Host A will disconnect.

I can't figure out why this is suddenly happening. This is the *same* vCenter server I have always used to manage this host, but about a month ago it started disconnecting every time I tried to manage it. In the vShpere client, I right-clicked and disconnected Host A and removed it from the list; then re-added it. But it will still disconnect after just a few seconds.  When it disconnects it is still visible in the vSphere client; it's just greyed out. I can right-click and select Connect. It will always reconnect but after a few seconds it will disconnect again.

I have rebooted all 4 hosts.
(The vCenter Servers, by the way, are VMs running on 2 of the hosts)

OK so Host A does not like to stay connected to vCenter Server #1.

As a test, I added Host A to the other vCenter Server (#2).
It works.
It stays connected to this Server and I can manage it just fine.

So now on vCenter Server #1 I only have Host B.
And now on vCenter Server #2 I have Host A, Host C and Host D.

This combination works fine, meaning Host A stays connected to vCenter Server #2.

As another test, I removed Host C and Host D from vCenter Server #2 and added them to #1.

So now on vCenter Server #1 I have Host B, Host C and Host D.
And now on vCenter Server #2 I have Host A

This combination also works fine.

It is only Host A that is problematic. For whatever reason it won't stay connected to vCenter Server #1.
Why would this all of a sudden start happening?

I found a knowledgebase article on vmware.com that says you can add a key to vCenter Server Settings called "config.vpxd.heartbeat.notRespondingTimeout" which increases the timeout. I set the value to the recommended 120. I restarted the vCenter Servers (in fact, I rebooted all of the hosts) but that did not seem to help.

I've tried replacing the network cable that runs from the management port (on the back of the host running vCenter Server #1) to the switch. I even plugged it into a different port on the switch. So I think I can rule out switch or cable issues. And besides like I said, I can manage this host just fine when I connect to it directly and I can also manage it from the other vCenter Server.

Everything is on one local network, all running through one managed gigabit switch.

vCenter Server #1 (the one that I have always used for managing Host A but now recently decided it does not like to stay connected) is running vCenter Server Appliance version 5.5.0.30500 Build 4180648 on linux.  

vCenter Server #2 is running Windows Server 2008 R2 Standard with vCenter Server for Windows, version 5.1.0 Build 1882349.

Both vCenter Servers have been untouched for many many months (meaning, I didn't just recently update either one, so I am also ruling this out as a possible cause)

I am getting ready to upgrade to VMware Essentials 6.5 but before I throw too many variables into the mix, I want to figure out why Host A will not stay connected to vCenter Server #1.

(By the way ... the reason I have 2 vCenter Servers is because I am using VMware Essentials. The Essentials edition only allows you to manage 3 Hosts at a time, so I purchased 2 liceneses ... I manage 2 hosts in each vCenter)