If a person arrives at the synagogue late, after the congregation has recited Nakdishach, or if he prays slower than the Hazan and has not begun the Amida before Nakdishach, he should not begin the Amida unless he can catch up to the Hazan before Modim. There is a Misva to bow together with the congregation when the Hazan recites Modim in the repetition. In order to ensure that one fulfills this Misva, he should not begin the Amida unless he feels capable of catching up to the Hazan in time to join the congregation at Modim. Otherwise, he should wait until the Hazan reaches Modim, bow with the congregation, and then begin his recitation of the Amida.
There is an alternative solution in such a case if the individual figures that if he begins the Amida, he will reach the Beracha of "Magen Abraham" at the same time as the Hazan reaches Modim. Since one bows while reciting the Beracha of "Magen Abraham," he can fulfill the Misva of bowing with the congregation at Modim if he recites "Magen Abraham" as they recite Modim.
This Halacha is codified by Hacham David Yosef in his work Halacha Berura (p. 81; listen to audio recording for precise citation).
Summary: A person who comes late to the synagogue, after Nakdishach, should not begin the Amida unless he feels he can catch up to the Hazan in time for Modim, or he can reach "Magen Abraham" just as the Hazan reaches Modim.
