I tried to use the search function first, but I didn't really know which terms to search for to get the results I wanted.

What happens is this: When the dampers contact the strings either after a key is released, or after the damper pedal is released, it makes a thwangy sound.

Is this a common problem for an older piano (1960 Kimball spinet)? Diagnosis? Solution?

Thanks!

"Thwangy"? LOL!!

Sounds like old, crusty felt. Some try to needle to get a little more life out of them, I'd suggest replacement.

Before considering damper replacement (which I highly doubt would be considered with a spinet), I recommend having a technician come over and adjust the pedals and the dampers. The damper spoons may also need to be adjusted as they may be preventing the damper from fully dampening the strings.

Often times on pianos like this, the dampers are becoming hardened or, they have acquired a little bit of rust from the strings onto the felt of the dampers causing this "zinging" sound as they engage the strings upon release.

A pretty simple solution is to (DO NOT DO THIS YOURSELF HAVE A TECH DO IT) take a very fine bristled tooth brush --- cut the bristles in half and gently brush the parts of the dampers that engage the strings. This can be done without removing the action. Or, the action can be tilted backwards a tad to get at the dampers easier.

It doesn't take much and it will eliminate that sound if that is what it is.

Thanks Jerry! "Zing" is a better onomatopoeia than "twang" for the sounds it makes. The strings aren't rusty at all, so the dampers must have just hardened some. I'll get my tech to check it out when he tunes.
Thanks again!