As an update to the issue
@Hydrozilla was having above -
The site called was running behind CloudFlare, and switching to another end point that didn't go through CloudFlare fixed the issue.
As such, it appears that the
[Any] Veterans Only (v1.0, 2015-01-22) plugin was (at least, sometimes) triggering the CloudFlare protection due to a too strict security setting on the site.
By connecting through the Tor network in a browser, CloudFlare gives 403 Forbidden as the status code, not "HTTP status: 0" (e.g. 0) as the plugin reported. The reason to this difference still remains a mystery though, and may need further testing.