We also discovered this fibrous type nest contained in a blue tit nest box. After some sleuthing on Google, we found this to be remarkably similar to the Wax Moth nest. A dense, difficult to remove, fibrous (with wood-dust, of almost sawdust-like consistency, in between cells where the larvae had gnawed grooves into the surrounding wood) material. Wax moths generally attack bumblebee nests, feasting upon it and building their own nests (which are too dense for bumblebees to pass through to fend them off) in its place. So it may be that a former bumblebee nest was present in your bird boxes, especially if it's been left an especially long time.

Here is a snapshot of a wax moth nest with the larvae still present... http://www.animalphotos.me/moth/moth-wax_files/waxcater.jpg

I have the exact same insect nest that I removed from a blue tit box. I live in Brackley, Northamptonshire. UK! Unable so far to identify what made it!

Moved from ID Request.