A question about using ferrite chokes in a data-acquisition system
All channels simultaneously get a value of 0 – that's most definitely not EMI, or any other type of what I'd call "noise".

That's a problem with your sampling hardware. Now, this might be anything – from a driver bug to a hardware fault, to samples not being fetched before some buffer runs over to some problems with a on-board power supply, or, and that's my current guess: a faulty connector somewhere that randomly makes the "BNC box" power off.

Generally, if you really just need 6 channels, your bandwidths are this low and you don't calibrated voltage readings, but just relative values: Try replacing the PCI DAQ card with a simple multichannel sound card.

A single 16bit ADC that gets multiplexed across a few 4kS/s channels? That's really not that great – you can actually build that with something as cheap as an arduino. Still, it's probably more time- and cost-efficient to not build something like that yourself, but try with widely available DACs – of which soundcards are the cheapest ones, and perform surprisingly well.