I've tested this on both my laptop (OS X) and my desktop (Ubuntu) but not a Windows box. As @til_b says, the problem seems to be that the .dbf has a different name to the .shp and .shx files.
After renaming the file behaved much better.
I also suggest you dissolve the layer using Vector > Geoprocessing Tools > Dissolve toolset.
After dissolving I was able to add, delete and edit features.
