I'm not very experienced either, but I bought a small key lime last year, put it in a small pot, it bloomed in Spring and my 5 limes are ripening now as well. But the tree also bloomed about 8-10 weeks ago, so I have both young fruit and ripening fruit.
Don't know if that helps....

I believe it is determined by the cultivar AND the climate.

My Eureka and Pomona Sweet Lemons, Bearss Lime, Thornless Mexican Lime all have flowers, small fruit, and large fruit on the trees.

When I lived on the beach, I had a Eureka that sometimes had 4 flowerings per year unless it got really cold that year. It only froze once in the decade I lived there.

So it would seem that an ever-bearing will bloom even with fruit on it. Okay, cool. Then it would appear that mine is not such a bush. Works for me! Thanks for the experiences.

And once again, Nature shows just how much she LOVES proving me wrong. While checking on my plants today, something catches my attention out of the corner of my eye... flower buds forming on my Key Lime. If I had waited just two days, I wouldn't have had to ask this question!