The bluebirds build mostly out of dried grass and pine needles. They weave a very neat nest. The Chickadees will line the nest with moss, and then some type of fiber. Twigs sounds more like a House Wren. Can you post a picture?

House Wrens will build many "dummy nests". I do not let them nest in my boxes. They will toss baby Blues and Dees out of their nests. They are Native and protected but I don't want them around.
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Well, like Lisa and Kendra said, it's neither of those birds if it's made of twigs. Sounds like a pesky House Wren to me too. They will fill up every box you have with twigs. Go and check any other boxes. The Chickadees use a thick layer of green moss covered by soft fibers. Bluebirds use mostly pine straw. I have included pics of both that I took in my yard.

Carolina Chickadee Nest

Eastern Bluebird Nest

i get the mountain and western bluebirds here and their nests are mostly dry grass and hay or grass stems woven neatly together but will include bits of string or yarn or the soft stringy bark from the juniper trees. and usually lined with feathers

Yep, it looks like the first picture with the green moss and feathers. I bought a new chickadee birdhouse hoping they take up residence there, and it appears they have. However, they are still looking at the other one too. I'd like a little more bird diversity (specifically blue birds)!

Good for you timjc! Chickadees are so much fun to watch! If you are looking to attract nesting Bluebirds, be sure to check out the Sialis site (www.sialis.org) for very specific information about what type of housing to offer and where best to place it. It matters a lot. HTH~~Angie

Thanks bandjzmom. I had some early interest from bluebirds as they were sitting on top of the birdhouse in the AM but would get scared away by the chickadees. I haven't seen them in a couple of days now.