You can certainly split them, either now 'in the green' or after they have died down (in which case mark their position). I don't see any point in digging them up and then storing them before replanting. Just seems like an extra chore.

The only Texas bluebells I know of are Eustoma aka Lisianthus which are not a bulb. AFIK it is a biennial which is grown as an annual.Can you say which plant you mean by 'Texas bluebell'?

Is this because they are overcrowded?
Crowding is the most common cause of them not blooming,, but there are lots of possible reasons. Please see link below for a list.

I have a few established clumps that bloomed well last spring, but have very few buds this year. Due to the very early spring and hot temps in the spring of 2012, I'm suspecting #4 at the link below, which I had not experienced before.

10. Growing conditions the previous Spring may have been inhospitable - the reformation of the bulb was affected. (An early heat wave may have shut down bulb rebuilding before it was complete.

These are up next to the house, on the west side.

Sue

Here is a link that might be useful: Daffodils Not Blooming?per The American Daffodil Society

Maybe they aren't "Texas" blue bells--just plain blue bells. When I planted them they were bulbs/corms, and had feathery purple blue flowers (like a loose hyacinth).
The reason I want to dig them up now is to refurbish the entire area and plant other plants. Other daffodils and tulips are blooming in the same general area.
Thanks for all the info.

Sounds like Spanish bluebells, Hyacinthoides hispanica. Practically indestructible. You can move them around any time.

I'd just get rid of as many bluebells as you can now. Those things will take over. Every year I yank as many as I can out of my flower bed long before the foliage turns brown and it never kills them. The foliage just falls flat all over everything else I am growing there. When I plant something new in that bed I dig up and throw out a whole bunch more. They continue to take over. They were one of the few things that my mother-in-law planted here before we bought their farm and I hate them!

You first need to positively identify your bluebells. When you say fluffy and bulb, I think of Camassia. Could they be this?

Here is a link that might be useful: Camassia