Am I correct in assuming the layer references a feature service you're trying to edit? If the feature service is running behind a firewall, ArcGIS Online won't have access to it. Thus, you can't edit it. Some operations, like displaying the layer do work, because the URL to the service is accessible to the browser, which is also running behind the firewall.
I'm trying to edit my data that's publically accessible (although the .com is behind basic .NET authorization, which I enter in order to view the data), but I receive the "Layer 'layer' seems to be on an internal network and is not accessible to ArcGIS.com. Thus, editing will be disabled on this layer." I can see it through the internet without any issues. Feature access is enabled.
I added the map through ArcGIS online by providing the feature service layer URL.