My lime tree is the teacher, and I am the student. Whenever I think I know what is going on with it, I am proven wrong.
The tree I bought from the nursery was marked as a Persian lime, like the ones commonly seen in supermarkets, but I'm now convinced it is a Key lime. I was convinced it had been rendered infertile by a hard freeze in its second year, but it is now evident that I just needed to be patient, because it produced a bumper harvest last year.
I should have planted the tree where it would get more direct sunlight; it's a little too close to the big live oak. I think that one factor in its late maturation was that it needed to grow tall enough to get more sunlight. Key lilmes need more heat units to produce fruit than any other citrus variety. I'm not sure how much that will ease my mind during particularly hot summers.