So, say there was a king has 3 sons. They would be princes, correct? And if that king died and his firstborn took the throne, would his brothers still be considered princes? If that king had children, and then died would the uncles of the current king be counted as princes still? I understand this question likely differs between countries (and language issues, such as the difference between the words Prinz and Fürst in German, for example) but I am not sure I understand how the term "prince" worked for historical kingdoms.
The use of prince you are referring to is the direct sons of a ruling monarch. If said uncle was the son of a king, then theoretically they would still be a prince.
That said, these men often received other titles and often those took precedence. If you were a prince who was made a Duke, you'd likely style yourself Duke before prince.