I had this problem too — solved it by reading gymreading’s post above. I design with a mac, using Chrome. I generally use the Mac keyboard shortcuts for copy (apple + C) and paste (apple + v). When I copied the the site key from the Google Captcha page using my keyboard short cuts, it copied and pasted just fine into word press. For reasons I do not understand, when I did the same thing for the secret key, it did NOT copy the code. So I was pasting the last thing it did copy into the secret key field, without knowing it. Of course this was causing reCAPTCHA to fail.
The fix was to copy the secret key code using the “copy” command under the “Edit” menu with my mouse, and then pasting using the “paste” command under the “Edit” menu with my mouse.
I Don’t know why this was happening, but it’s fixed now and working just great. Another happy customer. Hope this helps someone!