Why does the Contact Form 7 plugin fail to send messages?

We've received reports that some sites that use the Contact Form 7 plugin may sometimes show an error message with an orange border, like these examples:

There was an error trying to send your message. Please try again later.

Failed to send message. Please try later or contact the administrator by another method.

This happens when Contact Form 7 thinks a message is spam. As that page explains, “Contact Form 7 has several different spam protection modules such as Akismet, reCAPTCHA, and disallowed list. To determine exactly which module has blocked the submission, it is recommended to use the spam logging feature.”

You may need to adjust the Contact Form 7 spam settings to avoid this problem.