Short answer: No, in most cases.
Long answer: Unlike metal fillings (the conventional kind also called mercury or amalgam fillings), white composite fillings are very technique sensitive. In order to be done properly, a white filling requires several steps that have to be done in the right order and with the correct timing. In between those steps, if the tooth is dried out too much, or one of the chemicals stays too long on the tooth, the result could be a very sensitive tooth long after the procedure is completed.
As always, there are exceptional cases and circumstances. Sometimes teeth with very deep cavities can be persistently sensitive for a few days or even a week or two, but this is not a rule and would still be uncommon. Also, some people are so exquisitely sensitive that they perceive pain even from the slightest stimulation. It is conceivable that these people experience pain more intensely and of longer duration after routine dental work than the average person.
If you’ve had several fillings done and they are painful for weeks or months afterward, this is not a typical response. I would encourage you to discuss the situation with your dentist or to seek a second opinion.