Problems getting Wake On LAN working on OptiPlex desktops

We recently got a number of OptiPlex 380 and 780 desktops.  I'm trying to get Wake On LAN working so we can use the power saving settings, but I'm running into trouble. 

WOL works when I'm testing, and putting the computers to sleep, etc, manually.  When I send a magic packet, the computers instantly come alive and I can connect to them.  If I try to wake the computer up after an hour or two, there is no response to the magic packet.  It appears something is shutting off the NIC, but I'm at a loss as to what.

The machines are slightly different, so I'll go with a 380 for now.  In the BIOS, I have "Low Power Mode" Disabled, "Remote Wake Up" is set to Enabled.  In Windows 7, on the network adapter Power Management tab, I have "Allow the computer to turn off this device to save power" disabled.  On the Advanced tab, "Wake Up Capabilities" is set to Both, and "WOL Speed" is set to "Lowest Speed Advertize".

Did I miss something?  Could something still be shutting down the onboard NIC?