We are using vSphere 5.5.0.
Using the vSphere Web Client, when I create a new VM or add a new NIC to an old VM, if I tick "connect" for the VM nentwork, the task is completed without error.
But when I reopen the settings window the checkmark is missing again and the VM is not connected to the Network.

The problem occours for every user, every VM on every host.
Those VMs are all Hardware Version 10 and can only be modified via the vSphere Web Client.
I created a new VM with Hardware Version 8, which can be edited with the vSphere Desktop Client.
I can connect this VM to the network using the Desktop Client, but can't do it via the Web Client.
It looks like a bug to me, but we have done no upgrades recently.