Related to rolling up the wire of the generator is a real experiment that went awry. The Tethered electric generator. The space shuttle had an wire they were going to lower. The spool jammed. The wire had to leave at a constant velocity. There was a lot of wire. Once the wire started to leave it couldn't stop. I suspect the design of the reel may have had problems similar to the problems suggested by John.
Part of the problem here is that it is assumed that the cable will be suspended from the airborne generator.
This limits its diameter and weight.
However, with the bottom end anchored to the earth and the upper end attached to a large mass in orbit above the geostationary orbit level, and you have it. The weight of the cable is not taken by the airborne generator. Indeed, using this principle, the generators can be suspended in a similar way which means they can be any weight and do not need to be capable off supporting themselves aloft.
There is an article here at Wiki http://en.wikipedia.org/wiki/Space_elevator
Of course, we still need a cable but now it has to carry a great deal more weight and I guess this is where we start to get into exotic materials.
Like most things, it is theoretically possible if we define theoretical as including some one magically coming up with the magical materials for the cable.
In reply to JMW, I can see a couple of problems in using the hypothetical 'space elevator' concept to provide support for the airborne generators.
As I understand it, the space elevator has to be located very near, not more than a few degrees of latitude, from the Earth's equator, so the electricity available from the airborne generators would be limited to the equatorial region. There was also a New Scientist news story a few years ago reporting that any trips into space using the space elevator would need to be 'agonisingly slow' (taking nearly a month) because the activity of climbing up the elevator itself might set up potentially hazardous oscillations. Therefore I can't see anybody wanting to attach high altitude wind turbines that are likely to produce destabilising sideways loadings to the elevator tower structure.