Its a hard one to answer because you have to define how hard you bite or pinch... you should also be aware that there is a risk (a small one but there all the same) that piercing can damage the nipples - especially if it gets infected or you have repeated piercings in the same place.

I would say that yes you can damage the nipples if you bite or pinch too hard. In theory you can bite the nipple right off if you bite hard enough and the human jaw can exert that much pressure. However, very few go that far. Use some common sense and build up your level of biting/pinching in slow stages. If at any point you get unusual sensations, colours or loss of sensation stop immediately and seek medical advice. However, for normal 'biting' it should not be a problem and since you are wary of nipple clamps because of the pain I suspect you are not biting that hard...