That is an unfortunately simplified view that we need to be careful about asserting.
As others have pointed out, the Old Testament books all precede the lifetime of Jesus, much less the founding of the Church. The New Testament books that ended up in the Christian canon were all written by the end of the first century. So from that point, the complete Scriptures were available.
On the other hand, it’s true that over the next couple centuries, additional works continued to be written that some folks claimed as Scripture, and not every Christian community held to the identical canon. It does seem to have been in the late 300s that local Church authorities started issuing lists of the canon that closely resemble our current Bibles. So in that sense, “the Bible” as we think of it now did not exist everywhere in that form until then, even though the inspired writings themselves had been around for centuries.
Heck, it took the Catholic Church as a worldwide entity until the sixteenth century, after the Reformation, to declare the canon in ecumenical council. So one could state that the canon wasn’t finally settled for Catholics until then, but we would hardly say there was no Bible before that.
Usagi