Sure, maybe in 500 to 1000 years... :) No, they don't age. Hydrometers can be damaged though. I am a metrologist, I used to calibrate things.
The little paper scale slipped into the stem can come loose; if it does (and that is exceedingly rare) you will develop an offset. If you take a very finely graduated scale and place it next to the scale on any hydrometer you will notice the scale is not linear, it's close but not exact, so if the scale moved it's not a good thing.
So here is how you check it. Look at the scales, somewhere on it it will tell you the calibration temperature. That is the temperature at which the scale is precise and can be read directly. If you can't find a temperature noted it's probably not very good to begin with but assume its 68F (20C). Get some distilled water at the calibration temp, stir it to make sure the temperature is equal and place the hydrometer in, spinning it as you lower it (like a top). It should read 1.0000 at the calibration temp; any deviation is considered an offset that has to be accounted for on all readings. An offset of
0.002 or less is not a big deal, just note it.
Cleaning can also affect it, the skin of the device needs to be very clean. I use Alconox to clean everything but that is not necessary, just use any good detergent you use now and dry it, you don't want water spots. If you want something perfectly clean you follow that with a rub down of grain alcohol, that leaves no residue. (It will look like you just wasted grain alcohol but believe me, it's clean now if it wasn't before.)
Temperature corrections are needed because water density changes with temperature and that is not linear either.
To calibrate a hydrometer used for fermentation pure sucrose is used. Table sugar is better than 99.5% pure so it's good enough. I have the tables for that and can email you them if you want to do that. In summary, temperature correction, cleaning and a check of zero are all that is needed for the most part.
Joe