In our generation it seems that Torah leadership is now inherited. Discounting the Hassidic world which has always done this, it seems now that the Yeshivish world is doing this as well. Note, I in no way mean to cast aspersions on any of the Roshei Yeshiva mentioned here, I am merely pointing out a trend.
If we look at many of the yeshivas we see that there is a dynasty.
Lakewood - Founded by R' Aharon Kotler, his descendents have been the Rosh Yeshiva
MTJ - founded by R' Moshe Feinstein, R' Dovid Feinstein is now the Rosh Yeshiva, R' Reuven Feinstein is the Rosh Yeshiva of Yeshiva of Staten Island (an offshoot of MTJ).
Philadelphia - the current Rosh yeshiva is R' Shmuel Kamentsky, R' Yaakov Kamenetsky's son.
Mir Yerushalayim - The founder was R' Nasan Tzvi Finkel, the current Rosh Yeshiva is also a Finkel
Brisk Yerushalayim - The founder was the Brisker Rav's son, the current Rosh Yeshiva is the Brisker Rav's grandson.
R' Shmuel Auerbach (R' Shlomo Zalman's son) is the Rosh Yeshiva of Maalos Hatorah.
You may ask, what is bad? The answer is that throughout history we have seen that many/most times sons of great men are not great. In fact this is a gemara. The gemara in נדרים פ"א states the following:
הזהרו בבני עניים שמהם תצא תורה ... ומפני מה אין מצויין ת"ח לצאת ת"ח מבניהן אמר ר יוסף שלא יאמרו תורה ירושה הוא אצלם ר ששת בריה דר אידי אומר כדי שלא יתגדרו על הציבור.
Be careful with the children of the poor because from them will come the talmidei chachamim ... R' Yosef said why is it that talmidei chachamim's son's are rarely talmidei chachamim? So that people will not say that Torah is an inheritance. R' Sheshes the son of R' Idi said so that they will not become haughty and imperious over the people (because they and their ancestors were Talmidei Chachamim).
We see from the gemara that Torah is not an inheritance and being the son of a talmid chacham not only doesn't guarantee that you will be one, but in fact the Gemara states that he probably will not be one.
If we look historically we will see this to be true. Look at the Rishonim, almost of all of their parents were not Rishonim (e.g. Rashi, Ramban, Rashba, Ran, Raavad, Ritva, etc.) and almost none of their children were Rishonim. The same goes for the Acharonim, who ever heard of the Vilna Gaon's son or the Ketzos's son. If we look at the pre-war Gedolim we see the same pattern. Who was the Chafetz Chayim's father, son? R' Aharon Kotler's, R' Yaakov Kamanetsky's, R' Shimon Shkop, etc.?
There have always been exceptions, but that is exactly the point they are exceptions. and they prove the rule.
This ties in to the previous post about kollel only schools. We see from here that this is not the way that things ever were in Klal Yisrael. As the gemara saysהזהרו בבני עניים שמהם תצא תורה, the next generation of Gedolei Torah will not necessarily come from the parents who are learning in Kollel.