The Time barrier was a theoretical limit in warp drive technology, which restricted the top speed of warp-powered vessels. The barrier was broken in the 23rd century.
In 2254, when the crew of the USS Enterprise discovered what they thought were survivors from the SS Columbia, which had crashed on Talos IV in 2236, José Tyler noted that the survivors would not believe how fast they could get back to Earth, due to the breaking of the time barrier that had occurred since the Columbia had crashed. For some time after the breakthrough, warp speeds were referred to as time warp factors. (TOS: "The Cage")
The warp speed limit for this barrier is not stated in "The Cage". However, according to ENT: "These Are the Voyages...", starships in 2161 were capable of warp 7 and, according to TOS: "Arena" and "The Changeling" (both set in 2267), while it was risky to sustain a speed of warp 7 or above for the Enterprise, the engine was capable of achieving warp 9.