I have the same problem, generally described as 'peeling' nails or 'brittle' nails though these two can be somewhat different: if the primary problem is that the layers of nail split in the plane of the nail, that is peeling (sometimes also referred to as splitting, though it is never clear which way from that term); if the nails break every which way in the plane and perpendicular to the plane, that is brittle. I have peeling nails. It doesn't really badly get in the way of polishing or growing the nails because when there is nail polish on them, it holds together the layers of nail preventing the peeling from temporarily being apparent, and the polish makes them more rigid, so the thinner end of the nail cannot break off.
My suggestion #1 is to regularly moisturize your hands and nails. Massage in the moisturizer (any kind, an oil is also fine) and pay attention to your cuticles and moisturizing above and under the tip of the nail and the full length of the finger.
#2 - Wear nail polish! This can be colour or clear coat. [Obviously with colour the mess at the tip of the nail is less apparent; but sometimes the colour will end up between the tip nail layers, where it cannot be removed until the nail grows out or the layer breaks off - just wear the same colour or other opaque colours in that case.] This might hold together the nails so that they can grow and gain length. If you are lucky with this regimen, you will get to the point where you have some length and can cut off the entire peeling end, possibly stopping the pattern. [This hasn't happened for me.] Even if not, the nails will not break as much while they are polished. Try to keep the nails in good polished condition, re-applying topcoat when needed, but avoid changing the polish until you have a chip. When it chips, fix it if that is possible by reapplying the polish on that nail. Do not leave the nail in "bad polished and chipped" condition as I find it can be vulnerable to more peeling at that point. Never start to peel off the polish without remover as this will certainly peel off a layer of nail, causing more damage.
#3 - Be super careful when removing the polish. Use remover; be gentle; move the cotton ball/swab only toward the tip of the nail to remove polish. When the nails are cleaned, take the time to moisturize your hands and nails for a few hours if you can before reapplying polish. (You will have to clean the surface of the nail before reapplying polish.)
#4 - As much as possible, do not slide a nail or tool between the layers to split off the broken layer. If it is still mostly holding together, leave it alone (you'll cover it with polish and it will help to keep the tip more rigid). If it is loose, use a file or clipper to remove only the loose part but do not file down the thickness of the nail anywhere else.
#5 - Some with this problem swear by (find very important) using a glass file. I've tried; it makes little difference for me. But no matter what file you use, file in one direction and finish with a fine file (or even buffing file/buffer) as it seems to help to seal the tip of the nail and layers together.
I've been doing this and in contrast to "no polish living" where my nails always stayed short due to breakage because of the thin end due to peeling, they can now grow. Some tips peel more than others, but they usually hold together when they are polished.
Good luck!