This question spins off of Why did Hagrid's umbrella work for him?, which discusses whether or not only Hagrid's wand, or indeed any broken wand, can be used.
Hagrid can clearly still use his wand, albeit only for simple spells, and even then, he does so only when Dumbledore gives permission. Ron's wand is broken, repaired with tape, and works, but poorly.
This conversation with Olivander implies that a broken wand might be useable, but shouldn't:
"Rubeus! Rubeus Hagrid! How nice to see you again.... Oak, sixteen inches, rather bendy, wasn't it." "It was, sir, yes," said Hagrid.
"Good wand, that one. But I suppose they snapped it in half when you got expelled." said Mr. Ollivander, suddenly stern.
"Er -- yes, they did, yes," said Hagrid, shuffling his feet. "I've still got the pieces, though," he added brightly.
"But you don't use them." said Mr. Ollivander sharply.
"Oh, no, sir." said Hagrid quickly. Harry noticed he gripped his pink umbrella very tightly as he spoke.
"Hmmm," said Mr. Ollivander, giving Hagrid a piercing look.
Philosopher's Stone - Chapter 5: Diagon Alley
Now it's not made clear that he's not to use them because it's not safe to do so, or if he's simply not allowed to.
In a military court martial (or at least in movies) a soldier is literally stripped of his insignia and emblems of rank. Now there's nothing stopping them from picking them up and sewing them back on, or buying new ones from the PX - the action is meant to signify that they are no longer in the army, and any attempt to present themselves as such will be met with harsh punishment. I've seen similar (more comedic) ceremonies in other professions in films - an accountant's visor is removed and his pencils broken, etc. The implication is "You are no longer one of us", not that the guy is rendered incapable of holding a pencil (unless the ceremony takes a very dark turn).
So the question is, is the breaking of a wizard's wand literally intended to render it inactive, or (my opinion) is it merely a symbolic ceremony intended to indicate that this person is no longer allowed to perform magic?