When a form generates an error indicating “There was an error processing the form”, that is a generic error the plugin gives for a bad nonce value.
What’s a nonce?
A nonce is a security device that (among other things) is used to prevent a form from being submitted remotely (such as from a spammer trying to hit the form handler and circumvent the form).
The WordPress Codex describes nonces this way:
A nonce is a “number used once” to help protect URLs and forms from certain types of misuse, malicious or otherwise. WordPress nonces aren’t numbers but are a hash made up of numbers and letters. Nor are they used only once, but have a limited “lifetime” after which they expire. During that time period, the same nonce will be generated for a given user in a given context. The nonce for that action will remain the same for that user until that nonce life cycle has completed.
It’s important to note the part about “[they] have a limited “lifetime” after which they expire.” Nonce values expire at a certain point and will then result in a failure (which is intentional, since that’s how it prevents someone from remotely submitting form data).
An example of this would be if you let a WP admin page sit open in your browser for a significant period of time and you click a link on that page and it results in an error page that indicates the link is expired. That’s from an invalid nonce.
What causes this issue
There is not one simple answer to what would cause an invalid nonce, but one of the most common is serving cached content for a form. If the nonce value is included as part of the cached page, it is likely expired and the form processing will fail, resulting in the error indicated above.
If you use a cache plugin, it is generally recommended to not cache pages with forms on them for this reason.