I recall reading in some other threads about some connectivity problems with different 82566 Gigabit Network Connections. In those cases the issue was with negotiating the proper speed. (See http://communities.intel.com/message/137080#137080 to read the thread.) Nevertheless, you might be better off using an older version of the driver that these people found helped them.
The network connection you have has been around for a few years without any connection issues. Therefore, going back to an older driver like the other people did might help you. The driver used by your network connection is used by many other devices, so I would not be too surprised to learn that an update that fixed an issue with one device broke something on another device.
So my recommendation would be to find the driver and software from whoever made your computer and use that version. Or you could try the version 16.3 software that worked for the people in the other thread. I found a copy of the version 16.3 package at http://downloadcenter.intel.com/Detail_Desc.aspx?agr=Y&DwnldID=18807.
I hope this helps.
I am sorry to hear that the driver change did not help. I suspect that your issue cannot be resolved by changing the driver.
Did you make any changes shortly before the connection issues started? Did you plug in something new to a USB port or add a new adapter to your system? Look for any changes that might be causing a problem and try reversing the change or rolling back to a Windows system restore point.
You might have some type of intermittent hardware error that is at the root of the problem. If the issue is with the network controller hardware, you could add in a new Ethernet adapter if you have an open slot. Or if you have a system that does not have any slots like a laptop, you could try a USB adapter.
Please post back about any resolution you find.