I ran into this problem where I was unable to install Firebug in Firefox. It would act like it was installing, Firefox would restart, but the plugin was simply not there.

A little research revealed there could be a variety of reasons for this, but for me it turned out Avast! antivirus was preventing it from working.

I disabled the Network service in Avast! long enough to try again, and it all worked.

Hopefully this will help someone else who runs into the problem.