According to our logs, the server returned a 302 redirect to some of your edit submissions. This is the correct response if the edit was made from the actual edit page (e.g. like this one), as in this case it's a standard POST/Redirect/GET. But since your edits are inline edits this should not happen; instead, you should receive a 200 response with a JSON object (inline edit submissions are AJAX requests). Trying to parse the response as JSON would thus fail, causing the error message.
To check whether an edit submission is inline or not (they hit the same route), we use .NET's
IsAjaxRequest(), which (so the Internet tells me) simply checks for the
X-Requested-With HTTP header which jQuery sets when making AJAX requests.
The fact that the server responds with a 302 therefore suggests that this header wasn't received. The obvious theory here is that the machine where these failures occur is sitting behind a proxy that strips this header, or something similar; maybe you could check that?