There's a bug in the contact form. When you send it always returns TRUE for the email send even when it's not sent.
$mail = wp_mail($emailTo, $subject, $body, $headers);
$emailSent = true;
}
if ($emailSent == true) {
$mail is set to the result of wp_mail which may be false, but the internal variable $emailSent is always true. $emailSent should be set to the value of $mail, probably.
There is also no error handling in place for a failed send.