Here is a simple way to get the bound 
  $$ a+b < (1.5+o(1))n. $$ 
Fix $\varepsilon>0$. If $a+b>(1.5+\varepsilon)n$, then either $\max\{a,b\}>n$, or $\min\{a,b\}>(0.5+\varepsilon)n$. In the former case $a!b!>n!$, hence $a!b!\mid n!$ is impossible. In the latter case, assuming that $n$ is sufficiently large, the interval $(0.5n,(0.5+\varepsilon)n)$ contains a prime. This prime divides both $a!$ and $b!$, whereas it enters $n!$ with the exponent just $1$ -- showing that $a!b!\mid n!$ cannot hold in this case, too.