Not completely following you...
But yes, you can ask too much from a woofer. Any Driver, really.
You can expect it to play lower than it physically can, and you can expect it to play higher. Both are deleterious to SQ.
Just in an academic sense:
If you have small 2-way standmounts, that perform well down to 60Hz, I would still cross those to a sub, even if just for music.
I would still follow the advice of crossing them around 100-120Hz. This takes power demand off a small woofer that is not designed to authoritatively perform strongly that low. (Sure it may sound good at reasonable volume levels, but lets consider that the LF performance still doesn't excel at even the bottom frequencies of Bass Guitar.)
Many of our friends here have agreed that a higher XO to Sub is advantagious to Mid Range and Treble SQ, while offloading the frequencies that demand more power to a specialized component.
I would still look for matching Sub and Speaker accordingly, in that I would want my Sub to perform well up to around 200Hz, even if I don't really use all of that, while at the same time searching for something that will at least cover the full range of piano. (Lets face it, you need a Big Sub if you are chasing Pipe Organ or other Infrasonics.)
Assuming a second order slope, a small 2-way should be well protected by offloading at 100Hz, and the same will go for the Sub. Though you won't be in danger of over-excursion at higher frequencies, some of those drivers just don't perform well at higher frequencies to to cone break up or whatever.
Hope that helps as I try to answer something I "think" you are asking.
If not, please detail exactly what's going on: what you are experiencing and your full equipment list. With that, we can better answer your question.
