When I turned sixty, I thought about Reb Yosef in Moed Kattan 28a. Reb Yosef used to give expression to his happiness by making parties for the Bnei Torah. When he turned sixty, he made a party to celebrate having passed the years during which his death might have implied the punishment of Kareis. (For examples of Reb Yosef's minhag, see Kiddushin 31 and BK 87a re: the obligation of the blind to do mitzvos and his talmid Abaye in Shabbos 118b re: a siyum, and, of course, our Gemara in Moed Kattan.) According to this Gemara, Kareis causes death before the end of the sixtieth year- in other words, before the sixtieth birthday, which is the first day of the sixty first year. The Gemara brings a passuk in Iyov (5:26) תבוא בכלח אלי קבר כעלות גדיש בעיתו, "You shall come to the grave at a ripe old age, as the grain stack is taken away in its time." The gematria of בכלח is sixty, so the Gemara darshens that death after the sixtieth year is not necessarily unnatural, and is therefore not evidence of some capital sin.
Reb Dovid Kohen of Gevul Yaavetz offers another source for this number. In parshas Breishis 6:3, it says "ויאמר ה' לא ידון רוחי באדם לעלם בשגם הוא בשר והיו ימיו מאה ועשרים שנה." Chazal tell us that this means that mankind had one hundred twenty years from that moment to do teshuva, and if they didn't, the mabul would come. However, some mefarshim understand that passuk as telling us that from that moment and on, humankind's years would diminish, until the natural lifespan would not exceed 120 years. (This is explicit in the Pirush HaRosh on the Chumash and the Ibn Ezra, and it can be read into the Chizkuni, and it can be seen in the newspapers every day.)
The passuk in Tehillim (55:24) says;"ואתה אלהים תורדם לבאר שחת אנשי דמים ומרמה לא יחצו ימיהם ואני אבטח בך" But You, O God, shall lower them to the Pit of Destruction; men of blood and deceit shall not live half their days, but I will trust in You., and Chazal (see Sanhedrin 69b and mefarshim on Bilaam in Pirkei Avos 5:19) understand this to mean that the wicked do not reach half their years.
If the maximum lifespan is 120, then reaching the sixty first year shows that the person is not a man of דמים ומרמה, and conversely, if he is a man of דמים ומרמה, he will not reach that age.
There is more to say about this. I just wanted to post it before it I put it away and forgot about it.