I tried talking with a few other experts to see if we could come up with any other ideas. One of them mentioned that they had run into something before where some of the other cards in the PCI bus were interfering with the board. In that case they removed all of the other PCI cards, inserted only the card that they were interested in testing and it worked. You might want to try doing that and seeing if you get similar behavior.
At this point, I'm not really sure what the problem is. There obviously is a conflict between that PCI bus and this particular board. The best workaround may end up being that you need to use a different motherboard.
We did find a solution to this problem. There were a couple of undocumented jumper settings on our motherboard. We found that through trial and error that a certain configuraton provided 3.3V to the secondary PCI slots. This allowed us to get the 6509 card working in a secondary slot, with the 6111 and PCI-GPIB cards installed in primary slots. We needed to re-install the NI-488 drivers in order to get the GPIB card to work though.