The big question is -- are all the failing sites hosted with the same company? I have a dozen sites that are running the latest WordPress and Contact-7 releases with no problems whatsoever.
The only ones that failed were hosted on servers that use the Qmail email system, as many hosting companies do... In every one of those cases, updating the class-phpmailer.php file to the version that will be included in WordPress 3.7 solved the problem. If problems still persist, it's most likely a plugin or template problem involving misuse of the current jQuery library installed by WordPress 3.6.
Contact-7 is not at fault. Either blame WordPress for releasing 3.6 with the previous version of class-phpmailer.php... or, the hosting company for running Qmail -- however, Qmail, a Unix based replacement for the older Unix sendmail, has many advantages over sendmail, especially superior security measures, which is why Qmail is more prevalent in serious security conscious hosting companies.