RESOLVED!  One of the test WordPress instances had an impossible, unpleasant and not covered on Google error message.  When user is trying to register, the CAPTCHA code is displayed properly and validation is correct, but when the button Register is press, the page would display this:

ERROR: Unreadable CAPTCHA cookie

I have a plugin “SI CAPTCHA Anti-Spam” installed.  The error message is from this plugin. I’ve tried changing browsers, clear cache and cookies. Nothing worked.  No luck on Internet search either.  What to do?

Finally I changed one of the settings on the plugin itself.  I check an option for:

Use CAPTCHA without PHP session

And it worked!  The error message disappeared.    Explanation for this option is:

 Sometimes the CAPTCHA code never validates because of a server problem with PHP session handling. If the CAPTCHA code never validates and does not work, you can enable this setting to use files for session.

More Errors

If you fixed this error, and now  getting a different CAPTCHA error, deactivate the plugin, delete plugin temp directory (captcha\temp), and activate the plugin again.
[2013-02-21 TH 17:28]

Additional WP Register page Observations – HTTP 500

If you are getting error HTTP 500, when user is trying to register, you need to change your SMTP settings on your EMAIL setups screen.  Apparently when SMPT server takes too long to respond to sent password and admin notification web server gets confused, and sends “Server not responsive” error.  I had to switch email SMTP settings from my local cable provider to one of the giants of search industry.