Thanks William and Rick. In that case, I will pot up those bulbils with roots (i.e. the one with leaves and the two off 'Soft Whisper') and see how they go.
I, too, concluded that there must be something causing so many lilies to produce bulbils. I had noticed the correlation between bulbil production and lack of blooms, although 'Tresor' breaks that pattern. What else (other than damage to the stem) can trigger bulbil production?