The point layers show up in the TOC in arcgis desktop, but they contain no features. However, some of the the point layers are fine (all or nothing). I'm sorry but I can not share the map as some of the information is confidential. But all I did was open the map in arcgis desktop 10.1 from arcgis online. The point features were created from uploaded zipped shp files. I would think that this problem is independent from the browser used, however I have tried Firefox, internet explorer, and google chrome.
I think that the problem is with the symbology. The points have links to images for the symbology. When this is changed to a generic cross, for example, the problem seams be gone.
I would like in the end to be able to have the complete map, with correct symbology, open in arcgis desktop.
Thanks for the feedback.