When I attempt to sync my X51, I get the following message: "The connection has limited or no connectivity. You might not be able to access the internet or some Network resources." I am unable to sync and the "green icon on the device just continues to run. Have I got some setting on my device not set right? Does anyone know how to correct this problem? Thanks for the help.

Are you trying to sync it through the USB cable/cradle or the BT/Wifi?
Try the USB connection. Also incase you have tried the USB already, check if the USB cable needs to be wiped clean and then reinsert in the port (trying a different port may help too)
With the Wifi/BT you may have to check settings of your connections on the PC.
Hope the above helps.
Best of Luck!

I don't have WiFi/BT on my desktop. Tried connecting with both the cradle and the sync cable. No luck either way.

Did you clean the contacts of the USB and try? The fact that the green AS icon mobilises means that it senses the Axim at the other end but can't get too connect it properly. This could be one of the Axim's lead contacts not matching the corresponding on the cable.
Also if that does not work: maybe you should do the thing we do with Windows programs: uninstall and reinstall AS!