I have been facing this issue in one of my client site. the CF7 suddenly stopped sending e-mails.
It was giving an error like the following.
"There was an error trying to send your message. Please try again later".
So, my first thought was anything wrong with the CF7 itself. I head back to the contact forms in the wp admin panel. In the mail tab of CF7, I checked every settings is OK. There is no validation error too.
The from e-mail address is also set as from the same domain.
Next my thought was there might be a hosting issue. Now, this site was hosted by siteground. It's been very rare they will block anything of CF7. I went to the cPanel. Checked the MX record. It was set to Google Mail System and the server is configured to use remote mail exchanger.
There were no problems too.
Then I went back to WP-Admin again and saw there is a post man plugin activated that is used to send mail from this site. I deactivated the plugin and BINGO!... the CF7 is working again.
I am telling you so much details so you know where to look up when you are in same kind of trouble.