I believe the city water department tests for a more detailed set of factors(I could not get proper information because I moved here recently, and I don’t yet speak Dutch). However, the tests I could buy/order from commercial labs around here only tests for specific ones (Lead, Mercury, Cadmium, Nickel, Coliform bacteria). I’m guessing that is because those were the common culprits.

By law, the water from the tap may only contain 1 microgram pollutants per liter, and the water companies usually set themselves stricter standards (and stricter than the standards for bottled water, IIRC). Assuming you drink 2 liters per day that is 0.066 grams over the course of a 80 year life. If the cause of the illness of your family is is the tap water, then it is very likely a problem with the plumbing in or near your house.

However, even that is very unlikely. Even if the plumbing in your house is lead, that's not likely to make you ill immediately (but it will over the long term). If the pipes leak, then the water is more likely to go out than pollution getting in, due to the water pressure. Given that several members of a family get ill, the overwhelmingly more likely explanation is either a contagious disease, or food poisoning.