Virtual Machines have lost their consoles

I'm using vSphere Client 5.5.0 to talk to a host running ESXi 5.5.

Today, I noticed that I can no longer access the consoles of the VMs.  I used to be able to, but now I cannot.

• There is no "open console" option in the "Commands" panel of the Summary tab
• If I select the "Console" tab, I get a black area that does not respond to mouse clicks at all

I've rebooted one of the "console-less" VMs, but it made no difference.

But I have created a new VM on the same host, and I can access the console on that machine as normal

Which makes me think that something has got ESXi confused. 

Can anyone shed any light on what it might be or how I can diagnose/fix this?

Robert.