When does semi-empirical mass formula break down? When does the semi-empirical mass formula stop working and what are its limitations? Is it when the difference in $N$ and $Z$ becomes large or when atoms become too big?  
 A: @farcher gave a nice link.
I just humbly attempt to give two points. And there is a natural question, why do we care, why should the formula be perfect?
1/ I think one of the limits is the shell structure of the nucleus. One can get it only from solving Schrodinger equation. Not from piling-up the nucleons on one heap and evaluating different kinds of energies. 
2/ the formula considers just the pronounced trends in the behavior of the nuclear matter. One can improve by further optimizing the parameters for different regions of the nuclear chart. But While this can lead to a better description, one would ask WHY? The model does not describe nuclear interactions behind this, so it's usefulness is limited. 
One can get some better models (like FRDM). What for?  E.g. they can be used to extract some features from systematics of experimental masses far from stability - something that could point to changes in the shell structure in cases we have nothing, but mass.
A: The semi-empiricle amass formula actually works better for heavier nuclei so the direct answer to you question evades me. Its very light nuclei where the discrepancy is large. I'm sure it was a typing error but you asked about failing when atoms became to large. I assume you meant nuclei instead of atoms?
