Worth trying, but not likely to be a problem of 2 machines on the same IP. I currently have 2 instances of gmail and 2 instances of "Elite Universe Mail" (gmail for companies) running. 1 of each in IE and FireFox.
You don't have to give up IE, but could take Firefox, Opera, or Chrome for a test drive.
Of course, there is the ultimate fix to 90% of all computer problems. Turn off everything, and reseat all of your network cables. Power up and try again.