The Elder Wand is an extremely powerful artifact that many a wizard would lie, cheat, steal and kill for. Dumbledore had hoped that he might die undefeated as it's owner, so that the wand's power would be broken. In the end of The Deathly Hallows, Harry too put it away in the hope that he will die undefeated and break the wand's power. For Harry this is an unlikely outcome, given his choice of professions.

But why did neither Dumbledore or Harry think of simply snapping the wand (as in the movie), and thus effectively destroy it? As we learn in The Deathly Hallows, a broken wand is no easy thing to repair. Harry's powerful phoenix feather wand could only be repaired by the power of the Elder Wand. A wand as powerful as the Elder Wand likely could not have been repaired, or if it could it probably would recover a mere shadow of it's former power.  

And on the off chance that someone could repair it to full potency, two or more pieces would be easier to hide more effectively than one whole one. Or in Harry's case he would have had knowledge of muggle methods that could have rendered it into dust or utterly burned it away.

So why did no one in the books think to physically destroy the wand, instead of trying to sit on it until death?