Note the photo and think about counting 5000 age rings on a pencil (roughly the situation with a core) and I think it becomes clear that 'recorded' is an important part of it.

As for accidents with trees, the Ténéré tree (https://en.m.wikipedia.org/wiki/Ténéré_Tree), an acacia tree that was standing alone in the middle of the Sahara, is worth mentioning.

In the story linked above they didn't know for sure, they just knew it was the oldest they had found yet. From that article:

Now, Currey almost certainly didn’t fell the oldest tree ever. There are forests in the White Mountains, and elsewhere, where trees currently standing are probably far older than his Prometheus tree. We just don’t know about them.

And then beneath that an update proving that he hadn't knocked down the oldest tree after all...

Update February 10, 2016: Since this article was written, an older tree was identified in the White Mountains, California. The tree is also a bristlecone pine and is thought to be over 5,000 years old.