Is it significant that my bike sometimes makes a "twang" sound when I hit a bump?
I would guess these are the spokes. When hitting a bump, you wheel gets compressed. The harder you hit it the more.

When you wheel is getting compressed some (Upper and lower) spokes obviously loose their tension. The wheel then will pop back in it's original shape due the rest of the spokes which are streched. This will bring back the tension to the compressed spokes which usually is noticeable with the "twang" noise. 

It may be a hint that you need to tighten your spokes. The looser your spokes are, the more compressed the wheel can get, which again can loosen your spokes even more. This can create a downwards spiral until the spokes start breaking. Simply check your spokes an tighten them if necessary.

Look I drew a nice picture :)