I'm sure it could, but we don't have any timeouts like that in the GUI, requests often take a minute or more to complete (the event polling), and we haven't seen reports of this behavior from anyone else, really... Since we don't have timeouts in the GUI there is nothing to increase. The refresh rate is "once" as it only loads the current status when the GUI loads and then polls for events, so there is nothing to change there either.
The dialog you refer to pops up when the GUI gets an HTTP or connection error from the other side. You should be able to see that in the debug console, and then we maybe could figure out the cause. Usually it's something like a proxy or antivirus, assuming again that it's not an actual error reported by Syncthing itself.