is it normal to

be very strong and not be able to do dips or pullups
in quantity?

Possible. You could have more fast twich fibers (strength/power) than slow twich (endurance). Define quantity...8, 12, 15 20? how have you trained in the past? If all you've ever done is 5X5 routines its no wonder you can't knock out 20 dips or pull ups.

Pullup are a tough exersise . Lots of people otherwise "in shape" have a hard time with them. You'e basically pulling you're weight (minus about 18% cause you dont lift your forearms and hands, but you get the point). Try pavel's "crease the grove" and see if it works for you.