This is known as cloning, and there are a few variables that need to be taken into consideration. The short answer is it depends.
Let's say you have a 20 gallon tank with 20 platys, cycled and running for quite some time. The filter media is capable of taking care of the bio load of 20 platys. Half the media is capable of taking care of the bio load of 10 platys.
So, you could take half the media, and half the platys, and set up another tank with half of the media & those 10 platys, and be reasonably sure there would be no cycling issues. Understand that nothing in the world is perfect, so fasting the fish for 24 hours before this transfer helps reduce the bio load slightly, leaving some room for error. This is cloning at its easiest.
A mature colony of nitrifying bacteria are capable of doubling every 24 hours. Mature does not mean cycled, it means running cycled for at least a few months. In a perfect world you could remove half of your bio media, and it will fully regrow in 24 hours. Sadly, we do not live in a perfect world.
The way around this? Remove no more than 1/3 of the bio media. Let's refer back to our 20 gallon tank with 20 platys.
1/3 of the media can support 1/3 of the fish, 6 or 7 platys. You can remove 1/3 of the media, put it in a new filter, on a new tank, and add 6-7 new fish, and reasonably expect no cycling problems on either tank. Once again, it helps to fast the donor tank for 24 hours, less food means less waste produced. This gives the media a little extra time to play catch up.
Few people have as easy a situation as our example tank, it is often more of a community tank. In these cases you are pretty much guesstimating the bio load in the donor tank, and starting a new tank with 1/3 of the media accordingly. With a close estimate, and keeping an eye on water parameters by testing there are usually few if any problems. Any spikes are usually minor, and short lived, resolving themselves in a few days with some extra water changes.
