The best way to increase pull up numbers imho

[Admin]

Rest:
Rest between sets if fine, I actually encourage it. You want to leverage rest so you can do the most work in the least time. If you can squeeze out 30 pull ups in a workout rather than 20 because you rested an extra 3 minutes total that is a big win. So if you can do level 6 rather than level 4 by addin a little rest by all means DO IT!

Failure:
Do not go to failure, stay 2 or 3 under failure, break up your sets.

Bottom line:
Do lots of small sets 2-3 under failure and you will see you numbers go up much faster.

Cheers

[samjackson]

I second Kevin's advice. While it goes against what we have learned in the various gyms we've been in, isolating muscles and working them to failure is counter productive to our goal of increasing functional strength.

Repetive, compound movements, not done to failure, with rest periods of 3 to 5 minutes between are what will yield dramatic strength increases.