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Sounds like an intermittent connection to the speaker winding.
You can try looking at the wires going to the cone in the back to see if anything looks broken.
May be difficult to fix unless you are a speaker expert.
You can re-cone speakers. http://www.simplyspeakers.com/speaker-recone-kits.html You do need to figure out what;s wrong. Is it the surround?
You can use a 1.5 V D battery to see if the cone moves freely in both directions. It can move on an angle because of the surround.
When the voice coil overheats, the speaker won't move at all or gets scratchy.
I think Crutschow has it right - this is not an unusual way for speakers to fail.
Check where the lead-in wires are soldered to the tags, that's the most common point for a failure.
The other end of the lead in wire often joins the voice coil propper under the dust cap, so if the break is there then you will have to cut off the dust cap (and then glue it back in position when done).
If that's all that's wrong then you should be able to do some kind of a repair, although it will probably never be quite like new. Considder piecing in a section of lead-in wire from a scrap speker to extend the length if you feel it may end up a bit short.
Failing that, yes - you may be able to re-cone it.
I take it that the cone is moving freely and it does not feel or sound like the coil is rubbing on the magnet?