Our Gemara on Amud Aleph and Beis tells us an interesting story about how Rav broke the news of his father’s death to his uncle, in an indirect manner:

When Rav went there, Eretz Yisrael, his uncle Rabbi Ḥiyya said to him: Is your father, Aivu, still alive?

⁦Not wishing to deliver unfortunate news, Rav said to him: Is my mother still alive? In other words, why do you not ask me about my mother, who is your sister, whether she is still among the living? 

Rabbi Ḥiyya said to him: Is your mother still alive? Rav said to him: Is my father still alive? Rabbi Ḥiyya understood from Rav’s failure to provide him with straight answers that both his brother and his sister had passed away. 

The idea of sensitivity when breaking bad news is also addressed later on on Daf 26b:

⁦The Sages taught the following baraisa: When a relative of a sick person dies, those around him do not inform him that this relative died, lest he lose control of his mind due to his emotional state and his grief exacerbate his physical health. And other people may not rend their garments in his presence, so that he will not know that one of his relatives passed away. 

Midrash Sefer HaYashar (Vayigash 9) brings down a similar idea in the way in which Serach bas Asher broke the good news to Yaakov about Yosef being still alive. She did so by way of a song so as not to shock him. We see here from our tradition that even good news can be harmful to an elderly or weak person who may not be able to bear the shock. 

Aside from common sense, there is scientific research that also supports this idea. According to research conducted by Dr. Ilan Wittstein of John Hopkins, there are indeed forms of heart attacks caused by shocking news. The medical term is stress cardiomyopathy. The interesting and key feature is that the influx of stress hormones, from an unusually shocking event, can actually lead to an interruption in the functioning of the heart and be as dangerous as a heart attack. Interestingly, the post “heart attack” treatment is different, because though at the time it is as deadly as a regular heart attack and requires immediate intervention to restart the heart, the period after does not require any invasive treatments. The person often recovers with no long term damage. It is important to be aware of this possibility, as otherwise doctors may misdiagnose it as a massive heart attack leading to extensive and unnecessary follow up treatments.

Source:

https://www.jhunewsletter.com/article/2005/03/sudden-shock-can-lead-to-heart-attack-81296