It only blooms on psuedobulbs that have not bloomed previously though they can send out a spike from each side of the pseudobulb and sometimes do not send them out at the same time. So, if a pseudobulb has bloomed from both sides, it will not bloom again.
Light might be an issue. When I had a Sharry Baby, I grew it in a large South-facing window with the Cattleyas and put it outside every summer on the edge of my porch where it received direct morning light and then very bright shade. You can also grow these under T5HO fluorescent lights if you do not have a good window.
You can certainly divide yours. I usually divided mine every year, in the spring, and they always bloomed for me. I think it best to keep three pseudobulbs per division and the best time to divide them is when the new growth is about to start roots.
I do not have a Sharry Baby any more as I wanted the room to grow other types of plants and smaller orchids...and I disliked the spots. I still have a Burr. Nelly Isler, though, which has the same basic growth pattern but doesn't seem as prone to spots or need as much light.
Onc. Sharry Baby- divide?
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