I have a problem of getting Summary Link Web Part (SLWP) work correctly for ages. There is no problem, if I add SLWP first to a page and add all links before I add a custom web part. However it doesn't work the other way, even adding a new link to SLWP is not possible after the custom web part is added.
The new links do appear in edit mode, but after save or publish the SLWP will remain the same. Edit the page again won't get the links back.
Any ideas?
It sounds to me like the custom web part is causing some issues on the page. I would download Fiddler2 and take a peek at the traffic between your browser and the server to see what's going on.
Start here: http://www.fiddler2.com/fiddler2/