The frame
On the Met's timeline for Egypt, 1000 BCE-1 CE, the Third Intermediate Period is dated "ca. 1070-712 B.C." and comprises Dynasties 21 to 24. The timeline describes it as a time of "competing power bases" with gradual fragmentation into smaller political units, many of them with origins in Libyan groups settled in the Delta in the late New Kingdom. The museum's essay on the period, by Marsha Hill and James P. Allen, divides it into four general stages: in the first, Dynasty 21 ("conventionally ca. 1070-945 B.C."), Egypt was governed by pharaohs ruling from Tanis in the eastern Delta and by the High Priests of Amun ruling from Thebes. According to the essay, the period began when, at the death of Ramesses XI in the early eleventh century BCE, the throne passed to Smendes. Sources 1, 2.
Boundaries and overlaps
At the end of the period two sources from the same museum contradict each other. The timeline ends it in 712 BCE and starts the Late Period, "Dynasties 25-30", in the same year; the essay is titled "Egypt in the Third Intermediate Period (ca. 1070-664 B.C.)" and calls Dynasty 25, "the Kushite Period", "the fourth phase of the Third Intermediate Period". The timeline itself dates "Nubian control, Dynasty 25" to ca. 743-664 BCE, so the dynasty crosses its own line at 712. Sources 1, 2.
Why no single year decides
Even within the essay there is no single end year: according to it, an Assyrian invasion involving the sack of Thebes, "ending in 663 B.C.", terminated Nubian control, while Tanutamani, the last pharaoh of Dynasty 25, is dated 664-653 BCE and retreated to Napata. The essay's phrase "some 400 years" of divided rule fits an end in 664 better than one in 712 (1070 to 664 is 406 years; 1070 to 712, 358). The timeline, for its part, ties 712 to no named event in the text read. The site's navigation uses the timeline's line at 712 so that Egypt's periods follow one sequence, and this page and the Late Period page state the essay's 664 explicitly. An object or article from Dynasty 25 is placed by the wording of its own source, not by a single year. Sources 1, 2.
On this site
"Tanis and Thebes: kings beside high priests" deals with the first stage, Dynasty 21, and with what the essay says, and does not say, about the two centers of power. "The Libyan dynasty and the fragmentation: Egypt after 945 BCE" continues from there: the family of Libyan descent, the weakening of its control, and the local rulers the essay lists for the end of the eighth century BCE.