When we break a magnet into two, both the pieces do have N and S poles. However it does NOT hold if we continue the process till we reach the atomic level. You can break a magnet down to a few thousand atoms or so and it'll still be a magnet, but beyond that it will just be a chunk of metal.
Magnets contain atoms that have their own little magnetic moment, or their north and south poles.In a typical magnet, these atoms align their poles so they point in the same direction. This adds up all of the magnetic energy inside the magnet.
Now this alignment of magnetic poles only travels so far throughout the magnet. Let's say you have a chunk of magnet that is 1,000,000,000 (billion) atoms across. Well, it turns out that there might be only a block of atoms that are 10,000 across that are actually aligned almost perfectly. We call that volume of atoms a magnetic domain.
The atoms can align their magnetic moments due to a very special interaction that takes place on the quantum level called the exchange interaction. Not only that, but the moments are also aligned on a very specific axis of the magnetic crystal, generally down a specific row of atoms called the "easy axis". It takes energy to pull those moments away from that direction, but the downside is that this energy decreases with the volume of your particle. In other words, if you shrink your magnet too much, this energy that holds the moments in place decreases below that of thermal energy at room temperature. Your magnetic poles then begin to flip like crazy, North switches to South, and back to North, quicker than you can blink. At this point, it's no longer a permanent magnet! It's at this point, the "superparamagnetic limit", where your particle's magnetic moment is diminished for all practical purposes. And that's the answer- no, you can't cut up a magnet down to a single atom. It stops being a permanent magnet a few hundred atoms before it reaches that point, depending on the specific alloy your material is made of.