Also, while doing this - push a bit on the wire -into- the connector from behind. This will slightly separate the contacts tangs from the plastic body by a hair and tends to make it easier to push the extraction tool over the tangs (less chance for them to bind up on the backside of the connector cavity).

I regularly stock up to 20 pin sizes for 0.156" and can get them up to 24 pins.
For 0.1" connectors, I have sizes up to 25 pins but have run out of 28s. Went through 15,000 of these (where the heck did they all go?!?!) and can't get this size anylonger. Even tried to get them custom made but no dice.