Some inequalities resulting from the fact that a binomial coefficient divides another binomial coefficient

I need to know some inequalities resulting from the fact that a binomial coefficient

${n \choose k} $ divides another binomial coefficient ${m \choose j} $. My searching returned no result during several days.

Is it true that we still have in general $n<m$. What are the conditions about $n,k,m,j$ to get for example $n≤m-2$.