Typically during a home inspection the water is only tested to see if it is safe to drink. You could end up with a skunk of a well, high in sulfates and iron making it virtually non potable or unable to wash your white clothes without rust staining them but still "safe" to consume.
I'm not sure if the home inspector can get further tests done, maybe the local ag department can help. What ever you do don't depend on one of the local water filter stores like Culigan because they don't have a clue beyond trying to sell you chemicals or a filter to fix a problem that may or may not exist. .
I ended up with a skunk well at a previous house and ended up installing a 2k air charge water system so we could at least shower without smelling rotten egg sulfur, but still couldn't drink the crap.
The well is in a really shallow pressurized aquifer, depending on rainfall, how much water we use, orbit of the moon, my horoscope and the stock market it will smell one day and then nothing the next day..
We could have possibly avoided that aquifer in another section of the property cross contamination was a risk I wasn't willing to accept at $200 a foot for the drill.
Other wells in the neighborhood have a history of going dry during the late summer "drought" but the three houses in our cul-de-sac haven't run dry in 35 years.. Bad water is better than no water..
Its called a run test. Fill the tub for 20min. to check for any slow down in pressure. This is also a septic test to see if it can handle 20 min of water flow.