While specialization for humans is becoming more and more of the "norm," I think Heinlein said it best:
"A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects."
-- Robert Heinlein, Time Enough for Love
So, in that context, yes -- every normal human should, in some fashion, be able to program a computer or a web app or what have you. I believe you should always seek to bring the knowledge and abilities of those around you UP to your level and if they exceed you, great, hopefully they will return the favor. Idealist, aren't I?