Update from @nod_:
This issue is too many comments long. The original issue: "Drupal alerts "An AJAX HTTP request terminated abnormally" during normal site operation, confusing site visitors/editors" has been fixed in #2808789: Fix "An AJAX HTTP request terminated abnormally" alert after user has navigated away from the page.
You could also check https://www.drupal.org/project/ajax_error_behavior, where most of the other initiatives and patches on this issue are bundled there.
Problem/Motivation (as of #269)
The following error is frequently run into with Drupal. Per #215, it does not apply to D8.
An AJAX HTTP request terminated abnormally. Debugging information follows. Path: /system/ajax StatusText: ResponseText: ReadyState: 4
It's easy to run into. It happens on any browser with or with out extensions. This is because it's the standard way of handling AJAX error in Drupal. If you submit a form, while an AJAX request is processing you receive the error.
To reproduce the error, enable a free tagging text field in a taxonomy which has a couple terms. Get the Ajax going by typing in the first couple letters of popular terms. While the Ajax request is going, submit the form.
This error is in-fact the intended purpose of Drupal and Drupal is throwing this error in Drupal 7 in the misc/drupal.js file via the Drupal.ajax.prototype.error = function (response, uri) in conjunction with Drupal.ajaxError = function (xmlhttp, uri).
These alert boxes are a nuisance and something like console.log or watchdog would be a much better place to put these errors.
The solution landed on in #265 (and before too) accounts for the desire to still print the error when it is truly an error. And in all cases when the error isn't an error, give first preference to window.console and as a last resort, alert the user.
User interface changes
Get rid of a nasty AJAX error.
Data model changes
Have options to display ajax errors in console log or in alert.