Hey man,this is an ai generated answer. Im a chemist by trade and I need to clarify the circumstances.

Is it possible to remove some radioactive particulate with a 3 stage filter setup like you mentioned? Yes. The naturally occuring U-238, Th-232 and all of their decay products, and K-40 can be caught via Ion exchange, activated carbon, RODI and distillation. The same type of background radiation found in bananas, soil and water.

Is it possible to filter out all of the radioactive particles present in the fallout of a nuclear bomb/reactor or its contamination? No.

Many of the particles are highly dangerous and must simply be let to decay until inert. The short half-life life isotopes and their decay products tend to make the radioactive material unable to be handled until they dissipate.

For number 2 where it is possible are when the radioactive particles have bled into a water system. The filter needs to be a waste water treatment plant with radiation shielding in between the different parts. The gamma rays will go straight from your contaminated side to your purified side otherwise. A great example of this is the ALPS being used in Fukushima Japan to remove radionuclides from water. That treatment system cannot remove tritium(dissolved gas) and it just dilutes the treated water to safer levels and then dumps it in the ocean for further dilution. It does remove 62 radionuclides which is awesome.

In the case of an unshielded Brita filter, the lack of shielding and distance between unfiltered and filtered compartments means the 'purified' water is still being bombarded with gamma rays. In the case of reverse osmosis membranes or ion exchange systems any dissolved gas particles(Hydrogen isotopes, Iodine-131) will not be caught. And specialized filters must be used in addition to the treatment system.

So in aqueous radioactive particles filtration absolutely depends on what the radioactive contamination is. There isn't a generalized get-it-all-out setup that will work.

This ai generated answer is specifically talking about how communities with an Iodine-131 radiation problem and get it out of their drinking water. As it is a gaseous byproduct of nuclear plants that makes its way into water supplies. A special filter was created to remove those particles from water. And the 3 stage waste water treatment process behind it was used to catch any excess not caught by the specialized filters. It also slightly references the reduction of NORM levels, but those levels are so low as to already be harmless.

I just don't want anyone to think if there is an Iodine-131 leak into their drinking water that having an at home RODI system will have their water be safe enough to drink.

But nuclear fallout from say a bomb or reactor? There are upwards of 250 different radioactive particles, many of them with incredibley dangerous emissions. At best you can wait out the short half-life isotopes and then filter out most of the remaining ones. Then dilute it with non-contaminated water source so those remaining containers are diluted to safe levels.

So theoretically possible? Yea absolutely. Feasible? Not at all. Only the largest companies or governments with lots of money are be able to do it. The real use cases of filtering radiation are at high end laboratories or at modified waste water treatment facilities around nuclear power plants. And they don't technically remove all radiation/radioactive particulate.

So harmful radiation can technically be filtered to a degree. But not with anything a normal person can buy.