So to summarize the outcome so far:
Taking Shaf from 6 to 20 pullups in 6 weeks is not possible.
Steve, yes volume based training can lead to overuse injuries, so great care has to be taken to chose the correct loadings, to vary the grips position etc. but whom do I tell that... All in all I far prefer volume over intensity based training. Both have there place, both work, both have shortcomings, but I once did some agressive waveloading to up my weighted pullups (got to an easy triple with 45kg) which gave me major problems with my right elbow...
What do you think of plan B (more or less identical to Pierres high rep idea). You do have a power rack and strong bands, correct? You should try to slip one strong band over the two safety bars. I would start around hip heights, stand on the band and try to do pullups. To increase the load you could simply lower the safety bars say every week. How many pullups can you do with the assistance of a strong band which is attached to the rack in the described manner?