It's strange, if only one client is dropping, I'd guess it means it doesn't get access to a certain resource (like mem or hd). OTH, if it's always the same toon that starts to lag, that's unlikely.
Is anything else in the area of the window used by that toon? Try changing the position of the window to see if it's position related - maybe some other application is trying to do something in that part of the screen (ok, wild guess here).
Have you checked your performance meter to check if any resource was hit hard at that moment?

But honestly, if it's always the same window that starts to bug, I'd check if is toon or position related. If it's toon related clean up the cache and interface folder for that toon. If it's position related.. well, then you need to check what other program is causing the trouble (like the update message of a virus checker popping up from the tray, a RSS reader rollout, something like that).