According to the Water Facilities of Budapest (Budapesti Vízművek) they measure water quality in multiply points. They measure 211.000 times (!) a year, and they check plenty of particles, chemicals, minerals, and other data. So tapwater in Budapest is definitely drinkable.
Water quality is measured for physical, chemical, microbiological, bacterial and radiological content.
The mineral content of the tap water in Budapest is ideal for the body, containing optimal amount of ions. Not less than the needed, and not more than optimal.
Conclusion: The quality of the water in Budapest is well above the European average.
If you are a fanatic tea or coffee drinker, water might contain too much mineral for making your favourite beverage.
If you are concerned about environment, you should definitely drinking tap water instead of PET bottled water.