I then got a helper to pump the brake pedal while I observed the pipes. Sure enough, one of them bulged under pressure. On any car which is more than a few years old, the flexible pipes should be checked and preferably renewed. In this case, new pipes cured the spongy feel. Manuals will tell you to change the brake fluid at least once a year, which is good policy even if few people do it.
Brake fluid absorbs moisture from the atmosphere.
If you don't believe this, fill a jar with fluid and leave it uncovered overnight. In the morning it will have overflowed. Fluid is incompressible and comes in Dot 4 or Dot 5 specification. If moisture enters the system, vapour forms under heavy braking and stopping power deteriorates. Unless the fluid has been allowed to drop too low, bleeding is seldom necessary.