Interesting.
Add-ins run from the ArcGIS Assembly cache. The DLL and any data included with the Add-in are copied there from the .esriaddin container (which is a ZIP file). However, the Add-in config is not copied there -- apparently loaded directly from the container. So if something is not correct with the cache, ArcGIS will still create the buttons and toolbars based on the config XML, but won't be able to run their code.
Try checking that the cache has been created correctly, and that the permissions on the folders and Add-in files you find there are correct.
On Win7/Vista, the cache is located at:
C:\Users\[username]\AppData\Local\ESRI\Desktop10.0\AssemblyCache
The AssemblyCache folder is a hidden folder.
On XP, it will be under Documents and Settings instead of Users with a similar path such as:
C:\Documents and Settings\[username]\Application Data\ESRI\Desktop10.0\AssemblyCache