If you're on ArcGIS 10.0, Esri SP5 might have a fix (NIM076505 & maybe NIM079744).
In lieu of that, if the feature that is causing the issue can be identified, you can delete it, reconcile, redraw it, and reconcile again. If the feature can't be easiliy identified, using the Esri tools to reconcile by Object in favor of Target should get through the reconcile. It'll probably be more work to get through the conflicts since they likely won't be resolved how you want, but the session should be salvageable at least.
It sometimes happens when a network feature is moved, a line is connected between the old position and the new position, and the reconcile proccess tries to resolve things in such a way that the node is moved back to the previous position. When it does that, the intermediate line feature can go to zero length (start and end points become the same), which of course SDE does not like at all.
I think there are other scenarios that cause similar issues, but that's the one I've come across most often. If the error doesn't give you object ID's, then it's usualy difficult if not impossible to identify what specifically is having the issue, and that's when I go to reconciling using the Esri tools.
Thanks, Matt. This is very helpful.
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Are your users reconciling in favour of the child or parent version? Sometimes you can get lucky and the other alternate option will work.
You can try to run the Verify Network Connectivity command from the geometric network editing toolbar to identify the feature.Here's some links to Esri's online help describing this option: ArcGIS Help 10.1
The only issue is if you have large data, this can take a long time since it run on a selection set. So as Matt stated, if you don't have a lot of edits, it might be best to recreate the session.
Thanks, Danielle. This is very helpful.