Is it possible to catch net::ERR_BLOCKED_BY_CLIENT?

Unfortunately you cannot catch that error message specifically, but you can catch the error itself:

$.ajax({
  url: 'http://openx.net',
  dataType: 'json',
  success: function( data ) {
    console.log( "Success:", data);
  },
  error: function( data ) {
    console.log( "Error:", data);
  }
});

Obviously the example isn't requesting JSON, but you can see that it fails and calls the error handler.

These errors are fired by Chrome when, for instance, a plugin like Adblock (as you mentioned) cancels a request.