ARCHER: We managed to avoid a court martial, but they grounded us for three months.
T'POL: Still, the NX program continued.
ARCHER: Eventually. The Vulcans had us run every simulation they could think of for over a year before they finally admitted the engine would probably work. Eight months after that, Duvall broke Warp Three in the NX-Delta. Five years later we laid the keel for Enterprise. You know the rest.
In "Unexpected" (ENT), which is mid-2151, somewhere between early May ("Fight or Flight" [ENT]) and late July ("Civilization" [ENT]), Archer mentions that he has known Tucker—to whom we see him introduced in "First Flight" (ENT)—"for eight years," placing the latter sometime in 2143. Following Robinson and Archer's unauthorized flight in the NX-Beta, they were "grounded for three months," and the Vulcans made them run simulations "for over a year" before the project continued. It was "eight months after that" when Duvall broke Warp Three. Depending on whether the three months of Archer and Robinson's grounding and the twelve-plus months of simulations ran concurrently or consecutively (which isn't clear), it's more than twenty months, or perhaps even more than twenty-three, before Duvall's flight. That places it in 2145. It's "five years later" that NX-01's keel was laid.
2143: Warp 2 barrier broken by Robinson
2145: Warp 3 barrier broken by Duvall ("over a year" plus 8-11 months after the above)
2150: NX-01 keel laid ("5 years" after Duvall's flight)
So, Franklin hails from sometime between 2145 (before which Warp Three was the absolute limit for Earth ships) and 2151 (when NX-01 was finally completed, and thus capable of reaching Warp Four and beyond). Again, that's if following the same timeline as ENT, which we can't say for certain is necessarily the case.
