Have you tried using some older versions of the drivers? Some people have had luck using older versions that are available from the OEM or even rolling back to the Windows verion of the driver. My first thought was you might be able to find the drivers from Dell, but I only see Windows XP and Vista for your model at
According to the version information at http://www.intel.com/support/network/sb/cs-006333.htm, support for Windows 7 SP1 was added starting with version 15.8.1, so you probably would not want to go to an older version than that one.
Try the drivers at http://downloadcenter.intel.com/Detail_Desc.aspx?DwnldID=20708. The drivers from this page are the version 16.5 package. You can ignore the description on this download. The packages do include Windows 7 drivers for the Intel(R) 82566 Gigabit Network Connections.
A slighlty earlier version, v16.2, of these drivers is at http://downloadcenter.intel.com/Detail_Desc.aspx?DwnldID=20011.
Let me know if one of the older drivers solves the connection issues.