He DID say everything he said.
It is, of course, impossible for Lawrence P. "Yogi" Berra to say something he never said. And he must have said everything he did say. Both of these statements are, of course, a necessary consequence of logic.
Consider this proof. Suppose Yogi did say things he didn't say. But that means he said something and didn't say it. This is a contradiction and hence, by reductio ad absurdum, we have proven it is false that Yogi said things he didn't say.
Similarly assume Yogi did not say everything he said. Then there must be at least one thing he did not say and yet he said. So eins mehr (as Ludwig Wittgenstein might have said) we have a contradiction and can by reductio ad absurdum conclude it is false that Yogi did not say everything he said.
Well, that's settled.
On the other hand, whether Yogi said everything that people say he said, that's something else again.
What complicates the matter is that Yogi may have said some of things to which he is attributed, but that doesn't mean - nor did Yogi claim - he was the ultimate originator of the quotes. In fact, very few quotes are completely original - with the possible exception of "Awe is but the product of an uncomprehending mind", which as we know, has arisen from one of the deepest thinkers of the twentieth century.
That said, Yogi apparently did say quite a bit of what he was supposed to have said. For instance, regarding a certain restaurant both Yogi and his wife Carmen agreed that Yogi did indeed say "People don't go there anymore. It's too crowded." The only thing they couldn't agree on was exactly what restaurant it was.
But perhaps the best bit of Yogi's advice was a product of his sixty-five year marriage. Once a friend saw him standing in line to buy tickets at the Metropolitan Opera.
"Yogi," he said. "What are you doing here?"
Yogi replied in one word.
Behind the Yogi-isms: Those Said and Unsaid, Victor Mather and Katie Rogers, The New York Times, September 23, 2015.
"Carmen and Yogi Berra: A love affair for the ages", Craig Wolff, New Jersey Star Ledger, March 8, 2014.