According to (at least) Rav Ovadiah Yosef (based on his understanding of Maran in Shulchan Aruch), one is not allowed to answer amen (except first 5 of Kaddish), baruch Hu u'varuch Shemo, or the yimloch response in Kedusha, if he is in the middle of the Elohai Netzor paragraph of the Amidah (i.e. after the first Yihyu leratzon, but before the second).
The proof comes from what Maran says in Shulchan Aruch that one should "speed up" his personal bakashot (that he is saying in the middle of Elohai Netzor, which is allowed) if he hears Kedusha or Kaddish approaching (so he can answer them properly).
This begs the question: why is he allowed to make bakashot and not answer amen? If bakashot are more important, then why does he have to speed up? If amen is more important, then why can't he say that too?
And bonus question of interest: why can't he just say amen (or any other response) as part of his bakashot?