The troubleshooting pages refer to an ETIMEDOUT error, but this appears to be related to the ArcGIS Server counter:
If the ArcGIS Server test results in an ETIMEDOUT error, the ArcGIS Server log file is too large to read before the request times out. Back up and delete the ArcGIS Server logs using ArcGIS Server Manager to resolve the issue. The log level and log content for your ArcGIS Server should be checked for errors to determine why the log file was so large.
Does anyone know if this is also true for the Amazon counter? IS this caused by the Amazon logs being too large to read before the request times out?