I have an old solar panel from 1984 on my roof. It is no longer connected. Could it still be good?
Solar electric or solar thermal?
Solar thermals never really go bad; they just bleach out the dark color on the surface that absorbs the sun's rays or the insulation degrades and falls off, depending on the style of the panels. You can paint the panels with a flat black paint or replace the insulation and then just lay it on your front yard and see if it gets hot in the sun.
Watch out, because they can get very, very hot. If it heats up, and it doesn't leak (test with a hose and a plug), there should be no reason that it wouldn't work on the roof -- mounted correctly, of course.
Solar PV systems, on the other hand, can go bad, can short out, contacts can be broken, etc., so it would be best to talk to a PV company and see if they can check it out (for a small fee, of course).
For more information:
You should also read Ricky Cappe's Ask A Pro Q&A, "We're buying a house with 25-year-old solar panels. Should we have the system updated?"