This depends on the design of the filters in the crossover.
Filters are defined by a few key characteristics:
- filter type
- cutoff frequency: this is the frequency at which the output is attenuated by 3 dB.
- slope: this is how much attenuation is applied. This is usually specified as 'x dB/octave': when you're one octave below the cutoff frequency, the attenuation will be x db. For every additional octave, you get x dB of additional attenuation. So if the filter is 12 dB/octave, the response will be -12 dB at one octave below the cutoff, -24 dB at 2 octaves etc.
The top green line shows the filter attenuation. This is a 12 dB/oct high-pass filter with a cutoff frequency of 1 kHz.
as you can see, the filter is not abrupt. It's very difficult to create a filter that just cuts off everything below the cutoff frequency - any analog filter that does this will have audible side effects.
A filter that has a gradual rolloff as shown here will have fewer side effects, but it will allow loud signals below the cutoff frequency to be heard.
So, If your crossover has a 12 dB/oct highpass filter set to 40 Hz and you send a 20 Hz tone through the filter (20/40 Hz is one octave), the output will be at -12 dB compared to the input. Depending on the volume setting on the amplifier, this will still be audible.
If the filter is steeper (say, 48 dB/oct) the output would be much quieter and may not be audible.
If you set the 12 dB/oct filter to a higher frequency (say, 320 Hz), you now have 4 octaves, so 4*12=48 dB of attenuation.