PROBLEM: Your bike pump doesn't work as well as it used to, and some strokes don't blow any air at all.
SOLUTION: It's very likely that the plunger's washer has dried out. When that's the case, the fix is easy.
Simply unscrew the end of the pump (opposite the end with the air chuck) and pull out the plunger. If the cupped washer on its end is torn or broken, check at your local bike shop for a replacement.
If the washer is merely dry and stiff, massage in some bike grease until it's pliable again.
It's safest to use grease that doesn't contain petroleum, which can degrade the inner tube rubber if it gets inside. For the same reason, avoid using oil.
Wipe off excess grease, then slide the washer and plunger into the pump. Screw the end cap firmly. Also, make sure that the chuck on the other end is tight.
Now go add air to a tire. The pump should work as good as new again!